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VERTEX OPERATOR CONSTRUCTIONS

arXiv:hep-th/9212087v2 15 Dec 1992PARTITIONS,VERTEX OPERATOR CONSTRUCTIONSAND MULTI-COMPONENT KP EQUATIONSM.J. Bergvelt, A.P.E.

ten Kroode,Urbana, December 14th, 1992Urbana, Illinois and Amsterdam, the NetherlandsUrbana, December 14th, 1992Abstract. For every partition of a positive integer n in k parts and every point of aninfinite Grassmannian we obtain a solution of the k component differential-differenceKP hierarchy and a corresponding Baker function.

A partition of n also determines avertex operator construction of the fundamental representations of the infinite matrixalgebra gl∞and hence a τ function. We use these fundamental representations tostudy the Gauss decomposition in the infinite matrix group Gl∞and to express theBaker function in terms of τ-functions.

The reduction to loop algebras is discussed.1. Introduction.1.1 Infinite Grassmannians and Hirota equations.Sato discovered that the Kadomtsev–Petviashvili (KP) hierarchy of soliton equa-tions could be interpreted as the Pl¨ucker equations for the embedding of a certaininfinite Grassmannian in infinite dimensional projective space, see e.g.

[Sa1, Sa2].Let us first recall the finite dimensional situation. The Grassmannian Grj(Cn)consists of all j-dimensional subspaces W of the n-dimensional complex linear spaceCn.

Let {ei | i = 1, 2, . .

., n} be a basis for Cn and let Hj ∈Grj(Cn) be the subspacespanned by the first j basis vectors e1, e2, . .

. , ej.

The stabilizer in Gl(n, C) of Hjis the “parabolic” subgroup Pj consisting of invertible matrices X = P XabEab,with Xab = 0 if a > j and b ≤j. Here Eab is the elementary matrix with as onlynon zero entry a 1 on the (a, b)th place.

So Grj(Cn) can be identified with thehomogeneous space Gl(n, C)/Pj. Now this homogeneous space is projective, i.e.,admits an embedding into a projective space.

Explicitly, let ΛCn be the exterioralgebra generated by the basis elements ea of Cn and ΛjCn the degree j part,i.e., the linear span of elementary wedges ei1 ∧ei2 ∧· · · ∧eij. For W ∈Grj(Cn)with basis w1, w2, .

. ., wj we have the element w1 ∧w2 ∧· · · ∧wj which is up tomultiplication by a non zero scalar independent of the choice of basis.

This thendefines an embedding φj : Grj(Cn) →PΛjCn. (If V is a vector space PV denotesthe associated projective space.) The image of φj is the projectivization of theGl(n, C) orbit of the highest weight vector e1 ∧e2 ∧· · · ∧ej and is described by1991 Mathematics Subject Classification.

primary 58F07, secondary 17B67, 22E65.Tt bAMS T X

2M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992the following quadratic equations: if τ ∈ΛjCn, τ ̸= 0 then [τ], the line through τ,belongs to Im φj iffit satisfies the following equation:nXa=1ψ(a)τ ⊗ψ(a)∗τ = 0. (1.1.1)Here ψ(a) and ψ(a)∗, a = 1, .

. ., n are fermionic creation and annihilation operatorson ΛCn that act on elementary wedges byψ(a) · (ea1 ∧ea2 ∧· · · ∧eaj) = ea ∧ea1 ∧ea2 ∧· · · ∧eaj,ψ∗(a) · (ea1 ∧ea2 ∧· · · ∧eaj) =jXk=1(−1)k+1δaakea1 ∧ea2 ∧· · · ∧ˆeak ∧· · · ∧eaj.

(1.1.2)Here the hat ˆ denotes deletion. The equation (1.1.1) is one of the forms of thefamous Pl¨ucker equations, cf., [GH].The infinite dimensional situation relevant for soliton equations of KP-type isinitially very much the same as in finite dimensions: one considers a group G ofcertain invertible infinite matrices indexed by Z, a parabolic subgroup P and thehomogeneous space Gr = G/P, an infinite Grassmannian.

(We will be sketchyin this introduction about the precise definition of the infinite dimensional objectsG, P etc; there are various choices for them, corresponding to various classes ofsolutions of the hierarchies).The group G has a central extension 0 →C∗→ˆG →G →0. (Such an extensionalso occurs in the finite dimensional situation of Gl(n, C), but is there necessarilytrivial and is usually ignored).

There is an integrable highest weight representationLλ for ˆG, with λ an integral dominant weight, such that the lift ˆP of P stabilizes thehighest weight vector vλ of Lλ. Then the projectivized group orbit P( ˆG·vλ) ⊂PLλis isomorphic to G/P and so this construction gives a projective embedding ofG/P.

The representation Lλ can be realized explicitly as a homogeneous component(with respect to the grading by “charge”) of a “semi infinite wedge space” on whichfermionic creation and annihilation operators ψ(a) and ψ(a)∗, a ∈Z act by formulaeanalogous to (1.1.2). The image of G/P in PLλ is described by (1.1.1), but withnow the summation running over all integers.To obtain the KP hierarchy one next considers the principal Heisenberg subal-gebra ˆsprinc of the Lie algebra ˆg associated to the group ˆG and one proves thatLλ, now thought of as a module for ˆg, remains irreducible under the action ofthe subalgebra ˆsprinc.

By uniqueness of representations of Heisenberg algebras oneconcludes that Lλ is isomorphic to a polynomial algebra C[x1, x2, . .

.] in an infinitenumber of variables.

An element τ of this polynomial algebra corresponds to apoint of the group orbit ˆG · vλ precisely when it satisfies an infinite collection ofdifferential equations (Hirota equations) of the formP(∂/∂x)τ(x) · τ(x) := P(∂/∂y)τ(x + y)τ(x −y)|y=0 = 0,(1.1.3)for certain polynomials P. The equations (1.1.3) are a “bosonized” form of thefermionic Pl¨ucker equations (1.1.1).This then is the KP hierarchy in so calledHirota form and gives the defining equations for the projectivized group orbit

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.3One sees that the construction of the KP hierarchy depends essentially on thechoice of the principal Heisenberg algebra to obtain a concrete, bosonic, realizationof the representation Lλ.It is therefore natural to investigate what happens ifone focuses one’s attention to other Heisenberg subalgebras ˆs of ˆg, that, as is wellknown, give rise to other so called vertex operator constructions ([KaP2, Lep]). Ingeneral the representation Lλ will not remain irreducible under other Heisenbergalgebras but in our situation there is in the group ˆG a subgroup ˆT, the translationgroup, of elements that commute with ˆs in the representation Lλ and such that Lλremains irreducible under the action of the pair (ˆs, ˆT).Investigating examples (see for example [tKB], [KaW]) one quickly discovers thatone obtains from these other constructions of Lλ in much the same way as beforedefining equations for the group orbit, but the equations can have a rather differ-ent character; in particular one will find hierarchies that contain also difference,as opposed to just differential, equations.

For example the Toda lattice can beobtained in this way. The difference equations are “caused” by the occurrence ofthe translation group in the vertex operator construction sketched above.

Also thedifferential equations that one obtains for other Heisenberg algebras look ratherdifferent: in the simplest case one obtains the Davey-Stewartson equation in steadof the KP equation.In this paper we want to discuss the hierarchies of soliton equations relatedto certain vertex operator constructions of the central extension ˆg of the infinitematrix algebra. These constructions use Heisenberg algebras of ˆg obtained from allpossible Heisenberg algebras of the affine Lie algebras ˆgl(n, C), where we think ofˆgl(n, C) as a subalgebra of ˆg.1.2 Lax and zero curvature form.Until now in this introduction we have described soliton equations in Hirotaform, using the representation theory of a central extension of the infinite lineargroup.Other approaches to these equations are the Lax and zero curvature formalisms.Let us sketch how these approaches are related to the representation theoretic one.As we discussed before the main ingredient in the recipe for the construction of Hi-rota equations was the choice of Heisenberg system (ˆs, ˆT) consisting of a Heisenbergsubalgebra of ˆg and a translation subgroup ˆT of ˆG.

Now using the sequences0 →C →ˆg →g →0,0 →C∗→ˆG →G →0(1.2.1)we obtain a pair (s, T) consisting of a commutative subalgebra in g and a subgroupT in G that commutes with s. The decomposition ˆs = ˆs+ ⊕ˆs−⊕Cc in annihilationoperators, creation operators and central elements induces a decomposition s =s+ ⊕s−. Then one considers on G/P “continuous and discrete time flows” from thepair (γ+, T), where γ+ = exp(s+).

The compatibility or commutativity conditionsof these flows will then be the Lax or zero curvature equations (depending on howone sets things up).It is well known (at least for the KP hierarchy) that the Hirota type equationsare equivalent to the Lax or zero curvature equations. The main point of this equiv-alence is the connection between the so called Baker function, or wave function, andthe τ-function, well known from the Japanese literature (e.g., [DJMK]) and also,in the algebro geometric situation in Russian works (e g[Kr]) In the case of the

4M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992principal Heisenberg algebra this connection was given by [SeW, 5.14], by a “sim-ple but mystifying proof” in the words of [W1]. For the case of the homogeneousHeisenberg algebra [Di3] uses this connection to define the τ-function.It is the aim of this paper to give a representation theoretic derivation of theconnection between the τ-function and the wave function.

(We will use the termwave function instead of Baker function in the main part of the paper). In ourset-up this relation is derived from the observation that the wave function consists,essentially, of one or several columns in a lower triangular matrix in the definingrepresentation of G (for the principal Heisenberg algebra or in general, respectively).Furthermore one finds that one can calculate matrix elements of such matrices interms of the fundamental highest weight representations of the central extension ˆG.The explicit use of matrix elements of fundamental representations of Lie algebrasto solve integrable systems goes at least back to the Kostant’s solution of the finitenon-periodic Toda lattice, [Ko].Let us describe this last simple, but essential, step in a finite dimensional situ-ation.

Consider the group Gl(n, C) acting on the vector space H = Cn, with, asbefore, basis e1, . .

., en and Hj the subspace of Cn spanned by the first j basis vec-tors. We recall that any g ∈Gl(n, C) admits a Gauss-decomposition g = g−Pg+,with g−a lower triangular matrix with 1’s on the diagonal, g+ an invertible uppertriangular matrix and P a permutation matrix.

The permutation matrix is uniquelydetermined by g, but the g± are not, unless P = 1n. We say that g ∈Gl(n, C)belongs to the big cell for Hj if PHj = Hj.

In this case we can choose the factorsin a variant g = gj−Pg′+ of the Gauss-decomposition in a unique way such that Pis the permutation matrix from the regular Gauss decomposition and gj−is of theformgj−= 1n×n +Xr>js≤jgrsErs,(1.2.2)If we put gj+ = Pg′+ we have gj+Hj = Hj and we see that the j-dimensional subspaceW = gHj of Cn projects isomorphically to Hj by the natural projection. Now itis not too difficult to see that the matrix elements grs of gj−can all be calculatedusing the fundamental representation ΛjCn (cf.

[BaR¸a, ch. 3.11]): we havegrs = ⟨Ers · vj | gj−· vj⟩= ⟨Ers · vj | g · vj⟩/⟨vj | g · vj⟩,r = 1, 2, .

. ., n,s ≤j,(1.2.3)where vj = e1 ∧e2 · · · ∧ej and ⟨· | · ⟩is the canonical Hermitian form on ΛjCnsuch that ⟨vj | vj⟩= 1.

The denominator τj = ⟨vj | g · vj⟩of (1.2.3) is the finitedimensional analogue of the famous τ-function; if we writegj+ =AB0D,(1.2.4)with A of size j ×j, B of size j ×n−j and D of size n−j ×n−j, then τj = det(A).The most important property is that τj is nonzero iffg belongs to the big cell forHj iffthe projection from W to Hj is an isomorphism.Translating (1.2.3) to our infinite dimensional situation gives Lemma 5.5.1. Re-call that one can associate to every partition n of n into k parts a vertex operatorconstruction for the infinite matrix algebra, using the technology of “bosonized kcomponent fermion fields” (see e g[tKvdL] and references therein)For each of

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.5these constructions we find solutions to the k-component differential-difference KPhierarchy and we obtain in theorem 5.5.2. by a straightforward calculation therelation between the wave function and the τ-function for these hierarchies. In theliterature k-component KP hierarchies were introduced in [DJKM2] and studied in[UT, Di1, Di3].

However there apparently only the solutions related to the homo-geneous partition n = 1 + 1 · · · + 1 are considered and also the difference equationsin the hierarchies seem to be included only implicitly.It is for various reasons interesting to study more general partitions.Recall(from [SeW], say), that one can associate a solution of the KP hierarchy to algebro-geometric data, consisting of a Riemann surface X, a point p ∈X with localcoordinate z−1 at p, a line bundle, etc.If z happens to be the nth root of aglobal meromorphic function on X with only a pole at p we have a covering mapX →P1 with p as the n-tuple inverse image of the point ∞∈P1 and one obtainsa solution of the n-KdV hierarchy. The natural generalization ([AB1, AB2]) ofthis construction consists in considering n-fold coverings of P1 such that the pullback divisor of ∞∈P1 is of the form Pka=1 napa, for p1, .

. ., pk points on X andthe na positive integers.

This gives us a partition n of n and by choosing otherappropriate geometric data (line bundle, trivializations, etc) one finds a solution ofthe k-component KP hierarchy. In [LiMu] this construction is used to study theanalogue of the Schottky problem for Prym varieties.

In [McI] these type of solitonequations are studied in terms of flows on generalized Jacobians, see also [Pr].In section 5. we will spend quite some time discussing the fermionic translationoperators ˆQa, the translation group ˆT constructed from it and the relation withthe infinite Grassmannian.In the algebro-geometric language the operators ˆQacorrespond to tensoring the line bundle with a bundle with divisor pa.Another way in which the hierarchies related to arbitrary partitions might beof interest is the following. Recently there has been much renewed interest in theHamiltonian structure of soliton equations in relation to the so called W-algebras ofconformal field theory.

For instance in [FeFr] the Wn algebras are constructed usingvertex operator algebras and the (modified) n-KdV hierarchy corresponding to theprincipal partition of n. The Hamiltonian structure of the n-KdV hierarchy is thereobtained using a remarkable duality of W algebras. It seems reasonable to expectthat there exist for every partition of n (or more generally for every vertex operatorconstruction of affine Kac-Moody algebras) a related W algebra and that usingduality of W algebras one can obtain Hamiltonian structures for the correspondingsoliton hierarchies.

Much here remains to be worked out, but see [BdG, BdGH,dGHM].There are many papers on soliton equations, so we list only a few. Our mainsources have been the papers [DS, DJKM, SeW, KP].

For background and furtherreferences on soliton theory we refer to the books [AbS, Ca, N, Di2]. Infinite di-mensional Grassmannians and infinite dimensional Lie algebras are discussed in themonographs [PrS, Ka].

Hierarchies of soliton equations in Hirota bilinear form re-lated to Heisenberg algebras and vertex operator constructions have been discussedin [KaW].2. The infinite Grassmannian.2.1 Infinite matrix algebra and group.

Let C∞be the vector space over C withbasis ǫi, i ∈Z. Let gl∞be the Lie algebra over C with generators the elementarymatrices (of size ∞× ∞) Ei j ∈Z that have as only non zero matrix entry a 1

6M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992on the i, jth place. We say, as usual, that Eij is upper (lower) triangular if i ≤j(i ≥j).

We have a natural action of gl∞on C∞given byEij · ǫk = ǫiδjk. (2.1.1)The group corresponding to gl∞is Gl∞, consisting of infinite invertible matricesX = Pi,j∈Z XijEij such that only a finite number of the Xij −δij is nonzero.We will need in the sequel infinite linear combinations of the ǫi and the Eij.These don’t occur in C∞, gl∞and Gl∞and therefore we introduceH = {mXi=−∞ciǫi | ci ∈C, m ∈Z},gllf∞= {Xi,j∈ZcijEij | cij ̸= 0 for only a finite number of m = i −j > 0},Gllf∞= {Xi,j∈ZXijEij | X invertible, Xij ̸= 0 for only a finite numberof m = i −j > 0}.

(2.1.2)So the matrices we consider have only a finite number of non zero lower triangulardiagonals but are for the rest arbitrary. The Lie algebra gllf∞and the group Gllf∞acton H by extension of the action (2.1.1).

This definition ensures that the exponentialmap of a strictly upper triangular matrix in gllf∞is a well defined element of Gllf∞,which is the main use we will make of these infinite sums. To deal with matriceswith an infinite number of both upper and lower triangular diagonals, for instancein applications in algebraic geometry, one could use the analytical setup of [SeW]or of [ADKP].

We warn the reader that in the literature on the Sato Grassmannian(e.g., [Sa2, AdC, KNTY, Mu]) one allows, in effect, sums that are infinite preciselyin the opposite direction from our definition, e.g., infinite number of lower triangulardiagonals, but a finite number of upper triangular diagonals. In this approach onecannot define the exponential of an upper triangular matrix.

As these papers show,one can circumvent this technical problem, and one would, by following this path,obtain a larger infinite Grassmannian than we do and a wider class of (formal)solutions of the hierarchies we are going to construct. In this paper we prefer toavoid these technicalities, so as to be able to use later on (in chapter 5) the resultsof the representation theory of [tKvdL], which is set up just in the present context.2.2 Infinite Grassmannian and Gauss decompositions.

Define in H for everyinteger j a subspaceHj = {jXk=−∞ckǫk} ⊂H(2.2.1)We define the infinite Grassmannian Gr of H as the collection of subspaces W ofH of the form W = gHj, for g ∈Gllf∞and some j ∈Z.The Grassmannian that we have defined here corresponds, mutatis mutandis, towhat is called the polynomial Grassmannian in [SeW, PrS]). We will need just a fewfacts about our Grassmannian that can be conveniently derived from a factorizationof elements of the group Gllf∞as products of lower triangular, permutation andupper triangular matrices.

This so called Gauss decomposition will also play animportant rˆole in the construction of the soliton hierarchies in chapter 4

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.7To formulate the Gauss decomposition in our infinite dimensional context letSfw∞be the group of infinite permutation matrices of finite width. So P ∈Sfw∞iffP ∈Gllf∞has a finite number of non zero diagonals and each row and columncontains precisely one non zero entry, which is equal to 1.

A permutation matrixP ∈Sfw∞acts on C∞by Pǫi = ǫσP (i), where σP : Z →Z is a permutation. Thisgives us a bijection of Sfw∞with the permutations σ of Z such that there exist aninteger N such that |σ(i) −i| < N, for all i ∈Z.Lemma 2.2.1 (Gauss decomposition).

Every g ∈Gllf∞can be factorized asg = g−Pg+,g−, P, g+ ∈Gllf∞(2.2.2)where g−∈Gllf∞, respectively g+ ∈Gllf∞, is strictly lower, respectively upper tri-angular, i.e., g−= 1 + Pi>j gijEij, and g+ = Pi≤j gijEij, P ∈Sfw∞. In case ghappens to belong to Gl∞also the factors g± and P do.The proof, which is not essentially different from the finite dimensional case, isleft to the reader.

Note that, as in the finite dimensional situation, the permutationmatrix P is uniquely determined by g, but that the factors g± are not, unlessP = 1∞. We will need a variant of the Gauss decomposition (2.2.2) determined bythe choice of an integer j. Letglj∞+ = {X =XXrsErs | Xrs = 0 if r > j and s ≤j},glj∞−= {X =XXrsErs | Xrs = 0 if r ≤j or s > j}.

(2.2.3)There is a natural projection prW,j : W →Hj, given by prW,j(f) = Pi≤j fiǫi iff = Pmi=−∞fiǫi. We will say that an element W ∈Gr belongs to the Hj cell whenthe natural projection W →Hj is an isomorphism.Lemma 2.2.2 (Gauss decomposition adapted to Hj).

Let g ∈Gllf∞be suchthat W = gHj is in the Hj cell. Then there is a unique decomposition of g of theformg = gj−gj+withgj−= 1∞+ X,X ∈glj∞−;g+ · Hj = Hj.Proof.

By the Gauss decomposition we have g = g−Pg+ and since W belongs tothe Hj cell we have Pg+Hj = Hj. Now write for the minus component of the Gaussdecomposition g−= 1∞+ Pℓ>m(g−)ℓmEℓm.

Then define a matrixf j = 1∞+Xm<ℓ≤j(g−)ℓmEℓm +Xℓ>m>j(g−)ℓmEℓm. (2.2.4)This matrix is lower triangular with ones on the diagonal, so is invertible and wecan define a new decompositiong = gj−gj+,(2.2.5)where gj−= g−· (f j)−1, gj+ = f j · P · g+.

Then gj−is of the required form and alsogj+Hj = Hj.■

8M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992We say that the decomposition described by this lemma is adapted to Hj, or toj for short.These decompositions are all related by conjugation. Indeed, if Λ is the shiftmatrix Pℓ∈Z Eℓℓ+1 in Gllf∞, thenΛ−1EijΛ = Ei+1,j+1,(2.2.6)so we see thatglj∞+ = Λ−jgl0∞,+Λj,glj∞−= Λ−jgl0∞,−Λj.

(2.2.7)This will be used in section 7.In general an element f ∈W is an infinite sum f = Pmi=−∞fiǫi. We say thatan element f ∈W has finite order (in the negative direction) if there is an integers, called the order of f, such thatf =mXi=sfiǫi,fs ̸= 0.

(2.2.8)We denote by W fin the collection of finite order elements in W.We use the Gauss decomposition to introduce a canonical basis for W fin. LetW = gHj and let g = g−Pg+ be the Gauss decomposition as in the lemma.

Thenwe have W = g−PHj, since g+ is an automorphism of Hj. Now a basis for thefinite order part of Hj is given by {ǫi | i ≤j} and a basis for (PHj)fin is providedby {ǫσP (i) | i ≤j}, where σP is the permutation corresponding to P. Then, sinceg−is strictly lower triangular, we see that by taking linear combinations of thefinite order elements g−· ǫσP (i) we can obtain a canonical basis of W fin given byws = ǫs +Xi/∈SjPi>sciǫi,(2.2.9)where s runs over the set SjP = {σP (i) | i ≤j} of orders that occur in W, and wherefor each s we have (for our definition of the Grassmannian) a finite summation.

SjPis a set of integers that is obtained from Z≤j by deleting a finite number of elementsand adding a finite number of integers > j. So SjP contains all sufficiently smallintegers.

Note that if s is small enough ws = ǫs, since there are in g−only finitelymany diagonals below the main one.The natural projection prW,j : W →Hj has finite dimensional kernel and cok-ernel. This follows, for instance, easily from the remarks about the canonical ba-sis of W fin we just made.

So we can define the index of prW,j as ind(prW,j) =dim(ker(prW,j)) −dim(coker(prW,j)). We have also ind(prW,j) = #(SjP −Z≤j) −#(Z≤j −SjP ), so that the index depends only on the permutation matrix P oc-curring in the Gauss decomposition of g, where W = gHj.

The index of prW,0 isalso called the virtual dimension of W, written Virtdim(W). The Grassmanniandecomposes into disjoint components of fixed virtual dimension: Gr = ∪j∈ZGrjwith Grj = {W ∈Gr | Virtdim(W) = j}.

For instance Hj belongs to Grj.An element g of Gllf∞is said to belong to the big cell if it has a Gauss decompo-sition with the permutation matrix P the identity More generally we say that g

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.9belongs to the Hj cell if it has a Gauss decomposition with a permutation matrixP as middle factor, such that the corresponding permutation σP is a product oftwo (commuting) permutations σj, σ⊥jwith σj (σ⊥j ) leaving the sets of integers{i | i > j} ({i | i ≤j}) pointwise fixed. An element g belongs to the big cell iffitbelongs to the Hj cell for all j ∈Z.If g belongs to the Hj cell the corresponding element W = gHj can be written asW = g−Hj, and projects therefore isomorphically to Hj.

In other words g belongsto the Hj cell iffthe element W = gHj does. This argument also shows that thevirtual dimension of W is in this case j.

An element W ∈Gr that is in the Hj cellhas a particular simple canonical basis (for its finite order part W fin):ws = ǫs +Xi>jciǫi,s ≤j. (2.2.10)Again, only a finite number of ws differ from ǫs.3.

Partitions and associated Heisenberg systems.3.1 Relabeling associated to a partition. Fix an integer n > 1.

Let n = (n1 ≥n2 ≥· · · ≥nk > 0) be a partition of n into k parts, so that we have n = Pk1 na.We relabel the basis for C∞such that we haveC∞=kMa=1Mi∈ZCǫa(i),(3.1.1)with ǫa(i) = ǫj, where j = np + n1 + · · · + na−1 + q if i = nap + q and 1 ≤q ≤na.We call ǫa(i) the type n relabeling of ǫj. For all positive integers n the principalpartition n = n, i.e., into one part, leads to same, trivial, relabeling: ǫj = ǫ1(j).The relabeling of the basis for C∞induces a natural relabeling of the basis forgl∞: an infinite matrix is then thought to be build up out of n × n matrices, eachof which consists of blocks of size na × nb, 1 ≤a, b ≤k.

More explicitly we putEnap+q nbr+sab= Enp+n1+···+na−1+q,nr++n1+···+nb−1+s,(3.1.2)and we haveEijabǫc(ℓ) = ǫa(i)δjℓδbc. (3.1.3)The multiplication for the generators after relabeling reads:EijabEklcd = Eiladδbcδjk.

(3.1.4)We extend this relabeling process for vectors and matrices in the obvious way toH and gllf∞.3.2 The numbers rb(j). Fix an integer j and a partition n = (n1, n2, .

. .

, nk) ofn into k parts. We then can write j = np+n1+n2+· · ·+na−1+q, with 1 ≤q ≤na,so that ǫj corresponds to ǫa(i), i = nap + q, in the relabeling of section 3.1.

Weassociate to these data the numbers rb(j) defined by:rb(j) =nb(p + 1)nap + qn pb < ab = ab > a(3.2.1)

10M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992In the sequel we will usually have fixed j ∈Z and we then will write simply rb forrb(j). These numbers satisfy:j = r1(j) + r2(j) + · · · + rk(j),rb(j) + nbℓ= rb(j + ℓn),ℓ∈Z(3.2.2)If n = (n) then rb(j) = r1(j) = j.

Note that, for any partition, the numbers rb(0)are all zero, so the reader might wish to keep this simpler case in mind.The meaning of these numbers is the following: consider the natural ordering onthe basis elements of C∞: ǫℓ≤ǫj iffℓ≤j. Let ǫa(i) be the type n labeling of ǫj,so that j = np + n1 + · · · + na−1 + q and i = nap + q.

Then the ordering on therelabeled basis vectors is given byǫb(m) ≤ǫa(i) ⇐⇒m ≤rb(j). (3.2.3)Another way of saying this is: ǫb(rb) is the largest basis vector of type b that issmaller than ǫa(i) (or equal, in case b = a).

So for instance we have:Hj = {jXt=−∞ctǫt}= {kXb=1rbXs=−∞cbsǫb(s)}(3.2.4)Combining the second relation of (3.2.2) and (3.2.3) we find:ǫj−n < ǫb(rb(j)) ≤ǫj. (3.2.5)The ordering of the basis ǫa(i) determines which relabeled elementary matrices areupper triangular:Emiba is upper triangular ⇐⇒m ≤rb(j),(3.2.6)if ǫj corresponds to ǫa(i).3.3 Pre-Heisenberg system of a partition.

Define now shift matrices Λ+a , Λ−a ingllf∞byΛ+a =Xℓ∈ZEℓℓ+1aa,Λ−a =Xℓ∈ZEℓℓ−1aa. (3.3.1)They act on the standard basis of C∞byΛ+a ǫb(i) = ǫa(i −1)δab,Λ−a ǫb(i) = ǫa(i + 1)δab,1 ≤a ≤k.

(3.3.2)LetHn =kMa=1 "Mk>0C(Λ+a )k# M "Mk>0C(Λ−a )k#! (3.3.3)be the commutative subalgebra of gllf∞generated by the shift operators.

We refer toHn as the pre-Heisenberg algebra of type n, since in the universal central extensionof gllf the lift of Hn is indeed a Heisenberg algebra see section 5 1

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.11When n is a partition in more than one part we have besides the pre-Heisenbergalgebra another ingredient in the theory: the pre-translation group. We need somedefinitions.

We write H = ⊕ka=1Ha where Ha is the subspace of H that can bewritten using only ǫa(i), i ∈Z. Let 1a := Pi∈Z Eiiaa be the projection operatorH →Ha.

Define operators on H byQa =Xb̸=a1b + Λ−a ,a = 1, 2, . .

., k.(3.3.4)Then Qa is invertible and we haveQ−1a=Xb̸=a1b + Λ+a ,a = 1, 2, . .

., k.(3.3.5)Let R = ⊕k−1i=1 Zαi be the root lattice of the simple Lie algebra sl(k, C), withαi, i = 1, 2, . .

., k −1, the simple roots.We define a homomorphism from theadditive group R to a multiplicative Abelian subgroup in Gllf∞byαi 7→Tαi := Qi Q−1i+1,1 ≤i ≤k −1. (3.3.6)In particular α = Pki=1 diαi gets mapped to Tα := Qki=1 T diαi.

The image of R iscalled the pre-translation group (of type n) and is denoted by T n.One sees immediately that elements of Hn and T n commute. The pair (Hn, T n)will be called the pre-Heisenberg system of type n.4.

Multicomponent KP equations.4.1 Time evolution. Let n be a positive integer.We are going to associateto every partition n of n a collection of “continuous and discrete time flows” onthe infinite Grassmannian Gr, using the pre-Heisenberg system (Hn, T n) of theprevious section.

Let Γn be the subgroup of elements of Gllf∞of the formwn0 (t, α) = exp(Xi>0kXa=1tai (Λ+a )i) · Tα. (4.1.1)Here t = {tai ∈C | 1 ≤a ≤k, i > 0} are the “continuous time parameters” andα ∈R, where R, the root lattice of sl(k, C), is thought of as a “discrete timelattice”.

We will often identify the pair (t, α) with the element wn0 (t, α) ∈Γn. Theelements (4.1.1) satisfy, of course,wn0 (t, α + β) = wn0 (t, α)Tβ,(4.1.2)for all α, β ∈R.We define the action (“time flow of type n”) of wn0 (t, α) on the Grassmannian inthe following way: for W ∈Gr we putW(t, α) = wn0 (t, α)−1 · W = exp(−Xtai Λia) · T−α · W.(4.1.3)If W = g · Hj then we have W(t, α) = g(t, α) · Hj, whereg(t, α) = wn0 (t, α)−1 · g.(4.1.4)Note that different choices of n and n might give the same flow on the Grassman-nian.

For example we obtain the same flow if we take for any positive integer n theprincipal partition n = n of n into one part. This are the famous KP-flows.We denote by Γj,nWthe collection of points (t, α) in Γn such that W(t, α) =g(t α)H belongs to the big cell with respect to Hsee subsection 2 2

12M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 19924.2 Formal Laurent series and pseudo differential operators. The multicompo-nent KP equation that we are going to introduce consists of equations for a k × kmatrix function of a “spectral variable” z, which appear as follows.Fix, as always, a partition n of n into k parts.Denote by ea, 1 ≤a ≤kthe standard basis vector of Ck with a 1 on the ath place and 0 elsewhere.

Wethink of the ea as column vectors.Similarly denote by Eab, 1 ≤a, b ≤k theelementary matrix in gl(k, C) with a 1 on the (a, b)th place and 0 elsewhere. Let,as usual, C[[z]] be the integral domain of formal power series in the variable z andlet C((z)) be its quotient field, the field of formal Laurent series.

Denote then byH(k) = ⊕ka=1C((z))ea the space of k-component formal Laurent series. Let nown : H →H(k) be the linear isomorphism given byn(ǫa(i)) = z−iea,(4.2.1)For any linear map A : H →H we have an induced map A(k,n) : H(k) →H(k)given by A(k,n) = n ◦A ◦(n)−1.

When the partition n is clear from the contextwe write and A(k).For any W ∈Gr we will write W (k,n) (or simply W (k), if n is fixed) for theimage n(W) ⊂H(k). The image of Hj ⊂H isH(k,n)j= ⊕kb=1C[[z]]z−rbeb = {kXb=1∞Xj=−rbcbjzjeb},(4.2.2)with rb = rb(j) defined in 3.2.1.

In H the subspace Hj is related to the standardsubspace H0 by Hj = Λ−jH0, for Λ the shift matrix Pi∈Z Eii+1 in Gllf∞. SimilarlyH(k)j= diag(z−r1, z−r2, .

. ., z−rk)H(k)0(4.2.3)On H(k) we have a natural action of the formal loop algebra gl(k, C((z))).

Often,in practice, it happens that the image A(k) of an operator A : H →H ends uplying inside gl(k, C((z))). This is not the case for Λ(k), in general, but for example,for a = 1, 2, .

. ., k we have(Λ+a )(k) = zEaa,(Λ−a )(k) = z−1Eaa,Q(k)a= diag(z−δab),T (k)αi = diag(zδi+1j−δij),i = 1, 2, .

. ., k −1.

(4.2.4)To check the first relation, we note that Λ+a ǫb(i) = δabǫa(i −1), so the lineartransformation induced by on H(k) maps (ǫb(i)) = z−ieb to δabz(z−i)ea, i.e., thisinduced map is multiplication by the matrix zEaa. The other relations are alsoeasily checked.In particular the group element wn0 of (4.1.1), responsible for the time evolutionon Gr, corresponds to multiplication on H(k) byw0(z; t, α) := (wn0 )(k) = exp(XkXtai ziEaa) · T (k)α ,(4.2.5)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.13whereT (k)α=k−1Yi=1(T (k)i)di,(4.2.6a)if α = Pk−1i=1 diαi. When we think of the root lattice as a sub lattice of ⊕ka=1Zδa,with αi = δi −δi+1 and (δa|δb) = δab, then we findT (k)α= diag(z−(α|δ1), z−(α|δ2), .

. ., z−(α|δk)).

(4.2.6b)Recall the evolution group Γn of subsection 4.1 and the corresponding time flowson Gr. Of all the generators of one parameter subgroups of Γn we distinguish aparticular one and call this the generator of the x-flow: we define,∂= ∂∂x :=kXa=1∂1a,∂ia :=∂∂tai.

(4.2.7)(See [FNR] for discussion of this process of singling out a particular combinationof the times as “x”). We will often let this operator act from the right, in whichcase we write ←−∂=←−∂∂x.

This operator acts on w0(z; t, α) byw0(z; t, α) · ←−∂= w0(z; t, α) · z. (4.2.8)We will also consider formally the inverse of ←−∂, defined byw0(z; t, α) · ←−∂−1 = w0(z; t, α) · z−1.

(4.2.9)We will in the sequel have to consider k-component formal Laurent series of theformw0 · X(z),X(z) =kXa=1mXi=sziXiaea,(4.2.10)so that the vector X(z) is a Laurent polynomial in z. We can, in this situation,“trade in” every occurrence of a power of z in X(z) for the corresponding power of←−∂: if we write X as in (4.2.10) and define←−X =kXa=1mXi=s←−∂iXiaea,(4.2.11)then, clearly, w0 · X = w0 · ←−X .

This procedure introduces in the theory the (noncommutative) ring of matrix pseudo differential operators, which will play an im-portant role in the sequel.4.3 The wave function. We will now use the concepts introduced in the previoussections to define the wave function

14M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Definition 4.3.1. Fix an integer j, an element W = gHj in Gr and a partition nof n into k parts.The wave function of type (j, n) of W ∈Gr is the k × k matrix function, definedfor (t, α) ∈Γj,nW , obtained by juxtaposing the k columns with index r1(j), r2(j), .

. ., rk(j)(these numbers are defined in (3.2.1)) of a certain infinite matrix and applyingto each column the isomorphism n (defined in the previous section) to get a k-component formal Laurent series:wW (z; t, α) = n wn0 (z; t, α) · gj−(t, α) · (ǫ1(r1)ǫ2(r2) .

. .ǫk(rk))= w0(z; t, α) · n(gj−(t, α) · (ǫ1(r1)ǫ2(r2) .

. .

ǫk(rk))).Mostly we will write wW (t, α) or even wW for wW (z; t, α). Note that the elementsǫ1(r1), ǫ2(r2), .

. ., ǫk(rk) all belong to Hj, so the columns gj−(t, α) · ǫb(rb) belongto W(t, α) = wn0 (t, α)−1 · W, and hence the columns of the wave function wW allbelong to W (k).

Note furthermore that the columns gj−(t, α) · ǫb(rb) are of the formgj−(t, α) · ǫb(rb) = ǫb(rb) +kXc=1Xℓ>0gcbrc+ℓ,rb(t, α)ǫc(rc + ℓ). (4.3.1)This is so because gj−· ǫb(rb) = ǫb(rb) + X · ǫb(rb) with X ∈glj∞−.

Now X · ǫb(rb)consists of a linear combination of vectors ǫc(ℓ) that are larger than ǫj = ǫa(i) inthe ordering of section 4.2, as one sees using the explicit form 2.2.3 of glj∞−andthe inequality (4.2.4). Applying the isomorphism to H(k) shows then that thecolumns of the wave function are of the form(wW )b(t, α) = w0(z; t, α) · n(gj−(t, α)(ǫb(rb))),= w0(z; t, α) · z−rbeb +kXc=1Xℓ>0z−rc−ℓgcbrc+ℓ,rb(t, α)ec!,= w0(z; t, α) · diag(z−r1, z−r2, .

. ., z−rk) · eb +kXc=1Xℓ>0z−ℓgcbrc+ℓ,rb(t, α)ec!.

(4.3.2)The summation over ℓin (4.3.1-2) is finite, since in our case gj−contains only afinite number of non zero diagonals.The point of introducing this wave function is that its columns form a basis foran important class of elements of W (k) over a ring of differential operators. Indeed,fix a (t, α) ∈Γn and defineW (k)fin (t, α) = {f ∈W (k) | f(z) = w0(z; t, α) · kXb=1mbXℓ=sbzℓfbℓeb!}.

(4.3.3)It is clear that W (k)fin (t, α) does depend trivially on α ∈R and we will delete α here.We will often also, having fixed t, suppress the t dependence. The space W (k)fin isthe image of the space W (k)(t, α)fin, the finite order part of W (k)(t, α), under theisomorphism given by multiplication by w0(z; t, α).

Note that if (t, α), (t, α + β) ∈Γj,nW (so that wW (t, α) and wW (t, α+β) exist) the columns of wW (t, α), wW (t, α+β)and also of ∂(wW )(t, α), ∂(wW )(t, α + β) belong to W (k)fin .We have the following generalization of results of Drinfeld–Sokolov [DS], Segal–Wilson [SW]:

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.15Proposition 4.3.2. Fix (t, α) ∈Γj,nW , let W (k) be the image of W in the spaceof k-component functions H(k) and define W (k)finby (4.3.3).

Then W (k)finis a freerank k module over the ring C[←−∂], with basis the columns of wW (t, α).Moreexplicitly: there exists for every f(z) ∈W (k)fina unique k-component differentialoperator ←−P (f) = Pkb=1←−P b(f)eb, Pb(f) ∈C[←−∂], such thatf = wW (t, α) · ←−P (f).(4.3.4)Proof. Let ǫj correspond to ǫa(i) and let rb the numbers associated with j in section4.2.

Suppose that for f(z) ∈W (k)fin as in (4.3.3) fbmb ̸= 0. Then we call mb + rbthe b-order of f(z) and fbmb the leading b-coefficient.

If all fbℓare zero the b-orderis −∞. (This ordering comes from a refinement of the ordering (2.2.3) on W(t, α),via application of and multiplication by w0(z; t, α)).For example, the bth column of wW has b-order 0 and leading b-coefficient is 1,whereas its c-order for c ̸= b is strictly negative.

If f ∈W (k)fin then also f ·←−∂∈W (k)fin ,with the b-order of f ·←−∂bigger by 1 than that of f and with the leading b-coefficientunchanged, for all b = 1, 2, . .

., k.If (t, α) ∈Γj,nW , then we have W(t, α) = gj−(t, α)Hj, so that W(t, α) projectsisomorphically to Hj. Applying we see that W (k)(t, α) maps isomorphically toH(k)j.

In particular if f = w0 · X ∈W (k)fin , with X = Pkb=1Pmbj=sb zjfbjeb thenX = w−10· f ∈W (k)(t, α) and X maps isomorphically to H(k)jusing the projectionpr(k)W,j. Now if the b-order of f were negative for all b then all mb would be smallerthan −rb and, taking explicit form (4.2.2) of H(k)jinto account, X would projectto 0 ∈H(k)j.

Since this projection is an isomorphism X, and hence f, has to bezero. So we see that for f ∈W (k)fin at least one of the b-orders is non negative, unlessf = 0.Now, of course, the idea is, given f ∈W (k)fin , to reduce its orders by subtractionof terms (wW )b · ←−P , for suitable differential operator ←−P .

More precisely, let m =mb + rb be the total order of f, i.e., the maximum of the orders that occur, and letµ be the multiplicity of m, i.e., the number of components c for which the c-orderis equal to the total order. Then (wW )b · ←−P , with ←−P = (←−∂)mfbmb, has the sameb-order m = mb + rb and the same leading b-coefficient as f and the c-order, c ̸= b,is at most m−1.

Subtracting we obtain an element f −(wW )b ·←−P of W (k)fin of lowerorder in the b-component. If the multiplicity µ was 1 the total order of f −(wW )b·←−Pis strictly smaller than that of f. In case the multiplicity is larger than 1 the totalorder of f −(wW )b · ←−P will still be m, but the multiplicity is one smaller than thatof f. By repeating this process we reduce the multiplicity and the total order andwe can find k differential operators ←−P a such that ˜f := f(z)−Pka=1(wW )a ·←−P a hasits b-order, for all b, less than 0.

Then, as we argued before, ˜f itself must be zero,so f(z) = Pka=1(wW )a · ←−P a. It is easy to check that the differential operators ←−P aare unique.■4.4 Differential difference multi-component KP.

In this subsection we derive theequations satisfied by the wave function as a function of the continuous and discretetime variables

16M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992The wave function wW (t, α) of type j, n, defined whenever (t, α) ∈Γj,nW , can bewritten aswW (t, α) = w0(z; t, α) · ←−w W (t, α),(4.4.1)where ←−w W is a k ×k matrix pseudo differential operator, called the wave operator,of the form←−w W = diag(z−r1, . .

., z−rk) · (1k×k +Xi>0←−∂−iwi),(4.4.2)with the wi k × k matrices with entries in the ring of functions B and the numbersrb defined in (3.2.1). To see this use the explicit form (4.3.2) for the columns of thewave function and use (4.2.9) to trade in negative powers of z for powers of ←−∂−1.The wave operator will be used later on to define resolvents.Note that in (4.4.2), according to our definitions, the summation over i is finite.Also note that ←−w W is invertible as a formal pseudo differential operator (PDO),since it starts out with an invertible matrix and contains for the rest only negativepowers of ←−∂.

Note finally that if we had used g−, the component of the ordinaryGauss decomposition (proposition 2.2.1) instead of gj−, the component in the Gaussdecomposition adapted to j (Lemma 2.2.2), to define the wave function in (4.3.1),the first term in the expansion (4.4.2) would have been not the identity matrixbut rather more complicated. The choice we make here also allows us to calculatethe wave function in a rather straight forward manner in terms of the τ-function,see Theorem 5.5.2.

In this sense the decomposition introduced in Lemma 2.2.2 isadapted to j.There exist unique k × k matrix pseudo differential operators ←−Λ a, ←−T αi, suchthat∂1aw0 = w0←−Λ a,w0(z; t, α + αi) = w0(z; t, α)←−T αi. (4.4.3)Explicitly we have←−Λ a := ←−∂Eaa←−T αi := ←−∂Eii + ←−∂−1Ei+1i+1 +Xj̸=i,i+1Ejj.

(4.4.4)Now defineRa := ←−w −1W · ←−Λ a · ←−w W = ←−∂Eaa + [Eaa, w1] + O(←−∂−1),Uαi := ←−w −1W · ←−T −1αi · ←−w W = ←−w −1W ·←−∂−1Eii + ←−∂Ei+1i+1 +Xj̸=i,i+1Ejj· ←−w W . (4.4.5)We refer to Ra and Uαi as the pseudo differential resolvents and lattice resolventsassociated to (W, j, n) respectively.

(See [GD] for the concept of a resolvent. Thelattice resolvent was introduced in [BtK]).

Note that in both Ra and Uαi the first di-agonal factor diag(z−r1, z−r2, . .

., z−rk) of ←−w W in (4.4.2) cancels, so that resolventsand lattice resolvents have the same general form whichever Hj cell or partitioninto k parts we use

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.17The lattice resolvent Uαi is an invertible matrix pseudo differential operator. Wesay that an invertible k × k matrix pseudo differential operator A is in the big cellif it admits a decomposition A = A−A+, where A+ is an invertible k × k matrixdifferential operator and A−= 1k×k + O(←−∂−1).

Such a decomposition, if it exists,is unique. The resolvent Ra is in general not invertible.

In fact RaRb = 0 fora ̸= b. (When k = 1 Ra is invertible, as a monic scalar PDO).

We will denoteby (Rai)+ the differential operator part of the ith power of Ra and by (Rai)−theformal integral operator Rai −(Rai)+, and similar for other possibly non invertiblematrix pseudo differential operators. So the notation subscripts ± is not entirelyunambiguous, but the meaning is hopefully clear from the context.Proposition 4.4.2.

Let W ∈Grj and suppose that (t, α), (t, α + αi) belong toΓj,nW . Then the lattice resolvent Uαi is in the big cell and we have:∂ibwW = wW · (Rib)+,wW (t, α + αi) = wW (t, α) · (Uαi)−1+ .

(4.4.6)Proof. Until now we have defined the action of PDO’s with a finite number ofnegative powers of ∂on w0 and the action of differential operators on expressionsone obtains in this way.

We also need to define an action of arbitrary PDO’s onexpressions of the form (4.2.10): we putw0 · X(z) · ←−∂Eab = w0 · (z + ∂)X(z)Eab,w0 · X(z) · ←−∂−1Eab = w0 · (z + ∂)−1X(z)Eab,= w0 · z−1∞Xi=0(−z−1∂)iX(z)Eab. (4.4.7)Here we run into a little trouble: w0 is a power series in z and there is a priori noguarantee that the product in the last line of (4.4.7) makes sense.

The easiest way tocircumvent this problem is not to try to calculate this product and instead interpretw0 as an abstract free generator v0 of a module N of expressions v0 · Pm−∞Fiziover the matrix PDO’s with action given by (4.4.7) (with w0 replaced by v0). (cf.[DS]).

In the obvious way we also define differentiation with respect to the times tbℓon N. We identify then wW = w0 · ←−w W with the element vW = v0 · ←−w W and theproof of this Proposition takes place in the module N. It happens that for someelements of N, such as vW · (Rib)+, vW · (Uαi)−1+ , one can give an interpretation asa formal Laurent series; in particular we can interpret vW · (Rib)+, vW · (Uαi)−1+ asthe series wW · (Rib)+, wW · (Uαi)−1+ . This being understood we will in the sequeljust write w0 for v0.As we noted before, the columns of wW (t, α) belong to W (k) and in fact to W (k)fin .The same is true of ∂ibwW and of wW (t, α + αi).

So we can use proposition 4.3.2 toconclude that ∂ibwW and wW (t, α + αi) are of the form wW (t, α) · O, respectivelyw(t α) P with O P k × k matrix differential operators On the other hand we

18M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992have by (4.4.1) and (4.4.3):∂ibwW = w0 · (←−Λ ib←−w W + ∂ib←−w W )= wW · (←−w −1W←−Λ ib←−w W + ←−w −1W ∂ib←−w W )wW (t, α + αi) = w0(z; t, α) · (←−T αi←−w W (t, α + αi))= wW (t, α) · (←−w −1W (t, α) · ←−T αi · ←−w W (t, α) · ←−w W (t, α)−1 · ←−w W (t, α + αi)). (4.4.8)Now w0 and hence also wW is a free generator for the action of PDO’s, i.e., if fortwo PDO’s X, Y we have wW · X = wW · Y then X = Y .

This impliesO = ←−w −1W←−Λ ib←−w W + ←−w −1W ∂ib←−w W = Rib + ←−w −1W ∂ib←−w W ,P = ←−w −1W (t, α) · ←−T αi · ←−w W (t, α) · ←−w W (t, α)−1 · ←−w W (t, α + αi)= (Uαi)−1 · ←−w W (t, α)−1 · ←−w W (t, α + αi). (4.4.9)Note that ←−w −1W ∂ib←−w W is an operator containing only negative powers of ←−∂whileO is a differential operator.

This implies←−w −1W ∂ib←−w W = −(Rib)−(4.4.10)Similarly ←−w W (t, α)−1·←−w W (t, α+αi) is of the form 1k×k+O(←−∂−1) and P = P(α) isan invertible differential operator. By the same argument we see that the operatorP ′(α) such that wW (t, α −αi) = wW (t, α) · P ′ is an invertible differential operator.Since wW is a free generator we have P ′(α + αi)P(α) = 1, i.e., the inverse of P isalso a differential operator.

From this we see that Uαi belongs to the big cell and←−w W (t, α)−1 · ←−w W (t, α + αi) = (Uαi)−. (4.4.11)Combining (4.4.9), (4.4.10) and (4.4.11) proves the Proposition.■Definition 4.4.3.

Let L be a matrix PDO of the formL = ←−∂A + O(←−∂0)(4.4.12)for A a diagonal constant matrix with distinct non zero eigenvalues Aa. Let w(z)be a solution ofw · L = zAw,(4.4.13)and introduce the resolvent and lattice resolvent associated to L by (4.4.5) using wfor wW .

Then the k-component differential-difference KP hierarchy is the systemof deformation equations for L:∂ibL = [L, (Rib)+],L(t, α + αi) = (Uαi)+ · L(t, α) · (Uαi)−1+ . (4.4.14)Consider the k × k matrix pseudo differential operatorLW (t, α) = ←−w W (t, α)−1 · ←−∂A · ←−w W (t, α) =kXa=1AaRa,←−∂A + O(←−∂0)(4.4.15)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.19Then wW LW = zAwW and one finds, using that LW , Ra and Uαi commute, thatLW is a solution of the k-component KP hierarchy, if (t, α), (t, α + αi) belong toΓj,nW .The compatibility equations for (4.4.6) are the “zero curvature equations” thatare also useful:∂ℓb(Ram)+ = ∂ma (Rℓb)+ −[(Ram)+, (Rℓb)+],(4.4.16a)(Ra(t, α + αi)ℓ)+ = −∂ℓa(Uαi)+(Uαi)−1+ + (Uαi)+ · (Ra(t, α)ℓ)+ · (Uαi)−1+ . (4.4.16b)4.5 An example: the Davey-Stewartson-Toda system.

In this subsection we dis-cuss a few of the equations that follow from the equations (4.4.16).Our starting point is an element W of the Grassmannian Gr and the choice ofa partition n, defining a time flow W 7→W(t, α), see (4.1.3). The simplest caseis obtained by choosing for any positive integer n the principal partition n = ninto one part.

The resulting equations form, of course, the KP hierarchy, discussedextensively in the literature, (see e.g., [SeW]), with the discrete part (4.4.16b)missing in this case.The next simplest case occurs when we choose for any n > 0 a partition n =(n1, n2) into two parts. Now there will be a doubly infinite set of continuous timeparameters (ti1, ti2), with i > 0 while the discrete parameter α lives on the rankone root lattice of sl2 with generator α1: α = mα1, m ∈Z.We will indicatethe dependence on the discrete variable by a superscript: we write W m(t) forW(t, mα1), etc.

We introduce some new variablesx = 12(t11 + t21),¯x = 12(t11 −t21),¯t = 12(t12 + t22),t = 12(t12 −t22),(4.5.1a)and the following differential operators:∂= ∂11 + ∂12,¯∂= ∂11 −∂12,∂t = ∂21 −∂22. (4.5.1b)Let h =100−1.

We will need the following resolvents:Rm¯x (t) = Rm1 (t) −Rm2 (t) = ←−w m(t)−1(←−∂h)←−w m(t),Rmt (t) = Rm1 (t)2 −Rm2 (t)2 = ←−w m(t)−1(←−∂2h)←−w m(t). (4.5.2)The resolvents (4.5.2) are defined iffthe corresponding element W m(t) belongsto the big cell.Lemma 4.5.1.

Let W m(t) belong to the big cell. Then the resolvents (4.5.2) havethe expansionRm¯x = ←−∂h +0qm(t)rm(t)0+ O(←−∂−1),(4.5.3a)Rmt = ←−∂2h + ←−∂0qm(t)rm(t)0+ λm(t)12 + µm(t)h++012(∂−¯∂)qm(t)12(∂+ ¯∂)rm(t)0+ O(←−∂−1).

(4 5 3b)

20M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Define Qm = µm + 12qmrm. Then we have the following equations for qm, rm, Qm:∂tqm = −12(∂2 + ¯∂2)qm + (qm)2rm −2qmQm,(4.5.4.a)∂trm = 12(∂2 + ¯∂2)rm −(rm)2qm + 2rmQm,(4.5.4.b)(∂2 −¯∂2)Qm = ∂2(qmrm).

(4.5.4.c)The equations (4.5.4) form the Davey–Stewartson system ([DaS, SaA]).Proof. We write for the wave operator and its inverse←−w m(t) = 12 + ←−∂−1wm1 + ←−∂−2wm2 + .

. .

,←−w m(t)−1 = 12 + ←−∂−1vm1 + ←−∂−2vm2 + . .

. ,(4.5.5)where wmi , vmiare 2×2 matrices and where we have ignored the irrelevant diagonalfactor diag(z−r1, z−r2) (see the remark after (4.4.5)).

Then we havevm1 = −wm1 ,v2 = −wm2 + (wm1 )2. (4.5.6)If we writeRm¯x = ←−∂h + γm + O(←−∂−1),Rmt = ←−∂2h + ←−∂γm + δm + O(←−∂−1),(4.5.7)thenγm = [h, wm1 ],δm = [h, wm2 ] + wm1 [wm1 , h] −2∂wm1 h.(4.5.8)This shows that γm is off-diagonal, so that we can writeγm(t) =0qm(t)rm(t)0,(4.5.9)and this proves (4.5.3.a).

Next we consider the zero curvature equation[¯∂+ (Rm¯x )+, ∂t + (Rmt )+] = 0. (4.5.10)Equating the coefficients of powers of ←−∂to zero gives the equations[h, δm] = −¯∂γm −∂γmh,(4.5.11.a)0 = ¯∂δm −∂δm.h −∂tγm + ∂2γmh + ∂γm · γm + [γm, δm].

(4.5.11.b)The first equation shows that we can writeδm = λm(t)12 + µm(t)h +012(∂−¯∂)qm(t)1(∂+ ¯∂) m(t)(4.5.12)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.21which proves (4.5.3.b).Substituting the Ans¨atze (4.5.9, 4.5.12) in (4.5.11b) weobtain the system2¯∂λm −2∂µm + ∂(qmrm) = 0,−2∂λm + 2¯∂µm + ¯∂(qmrm) = 0,−(∂2 + ¯∂2)qm/2 −2µmqm −∂tqm = 0,(∂2 + ¯∂2)rm/2 + 2µmrm −∂trm = 0,(4.5.13)Now we eliminate the variable λm by imposing on the first two equations of (4.5.14)the integrability condition ¯∂∂(λm) = ∂¯∂(λm). This gives∂2(2µm −qmrm) = ¯∂2(2µm + qmrm).

(4.5.14)The last two equations of (4.5.13) combined with (4.5.14) and the definition of Qmgive then the Davey–Stewartson system (4.5.4a-c).■To derive the discrete equations we need the lattice resolventU m(t) = ←−w m(t)−1 ←−∂−100←−∂←−w m(t). (4.5.15)Lemma 4.5.2.

Assume that W m(t) and W m+1(t) belong to the big cell. Thenthe lattice resolvent (4.5.15) admits a factorization U m = U m−U m+ where U m−=12 + O(←−∂−1) andU m+ =02/rm−rm/2←−∂+ 12(∂−¯∂) log(rm).(4.5.16)Proof.

The fact that the factorization exists follows from the proof of Proposition4.4.2. To calculate the positive factor writeU m(t) = ←−∂E22 + [E22, wm1 ]++ ←−∂−1E11 −∂wm1 E22 + [E22, wm2 ] + wm1 [w1, E22]+ O(←−∂−2),(4.5.17)where we have used the expansion (4.3.6).

Parametrize wm1 =amqm/2−rm/2bm.Then we find from (4.5.9)δm = [h, wm2 ] + −12qmrm −2∂am∂qm −amqm∂rm −bmrm12qmrm + 2∂bm. (4.5.18)Combining this with the explicit form (4.5.13) of δm gives[h, wm2 ] =0−12(¯∂+ ∂)qm + amqm1(∂¯∂) m + bm m0(4.5.19)

22M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Since [E22, wm2 ] = −12[h, wm2 ] we find for the lattice resolvent the expansionU m = ←−∂E22+0−12qm−12rm0+←−∂−1 1 + 14qmrm14(¯∂−∂)qm14(∂−¯∂)rm−∂bm −14qmrm+O(←−∂−2). (4.5.20)A simple calculation shows then that U m+ has to have the form (4.5.16).■A straightforward but tedious calculation now shows that−¯∂U m+ (U m+ )−1+U m+ (Rm¯x )+(U m+ )−1 =←−∂−4/rmrm4 (¯∂2 −∂2) log(rm) −14qm(rm)2−←−∂(4.5.21)Hence, using (4.4.16b), we findqm+1 = −4/rm,(4.5.22a)rm+1 = rm4 ((¯∂2 −∂2) log(rm) −qmrm),(4.5.22.b)With um = log(rm) the equations (4.5.23) imply14(¯∂2 −∂2)um = exp(um+1 −um) −exp(um −um−1),(4.5.23)the 2-dimensional Toda lattice equation.From the diagonal components of theexpression −∂tU m+ (U m+ )−1 + U m+ (Rmt )+(U m+ )−1 we findλm+1 = λm + ∂¯∂log(rm),µm+1 = µm + 12(∂2 + ¯∂2) log(rm),(4.5.24)from which followsQm+1 = Qm + ∂2 log(rm).

(4.5.23.c)Note that, for fixed j, W m(t) will be outside the Hj cell for all t except possiblyfor a finite set of integers m. In fact, as in [BtK, Lemma 6.1.b], one can prove thatthis set is, in the case of the polynomial Grassmannian we use, an uninterruptedsequence mmin, mmin +1, mmin +2, . .

., mmax, so that we get from this constructionsolutions of the finite 2-dimensional Toda lattice equation. To obtain solutions ofthe infinite 2-dimensional Toda lattice one has to choose a suitable W belonging toa bigger space, e.g., the Segal-Wilson Grassmannian.5.

The calculation of the wave function from Λ∞2 C∞.5.1 Semi-infinite wedge space and Fermi-Bose correspondence.In this subsection we collect some results from [tKvdL] on the Fermi-Bose cor-respondence associated to a partition of n that we will need later, see also [DJMK,Ka].The space C∞is a representation for the group Gl∞, but not a highest weightrepresentation. The fundamental highest weight representations of Gl∞are con-tained in the semi-infinite wedge space Λ∞2 C∞, the collection of (finite) linear com-binations of semi-infinite exterior products of elements of C∞of the formv ∧v∧v∧(5 1 1)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.23where {vi}i≤0 is an admissible set: a set {vi}i≤0 of elements of H is called admissibleif there exist integers k ∈Z, N ≤0 such that vi = ǫk+i for all i ≤N. Note thatwe might as well assume in the construction of a semi-infinite wedge that the vibelong to C∞.

(This is the reason for the notation Λ∞2 C∞instead of Λ∞2 H.) Forlater constructions, however, we prefer to allow elements of H as members of anadmissible set. The integer k is called the charge of the wedge v0 ∧v−1 ∧v−2 ∧.

. .and the semi-infinite wedge space decomposes in a direct sum of subspaces of fixedcharge:Λ∞2 C∞=Mk∈ZΛ∞2k C∞.

(5.1.2)If in an admissible set {vi}i≤0 all the vi are of the form vi = ǫji then the corre-sponding elementǫj0 ∧ǫj−1 ∧ǫj−2 . .

. (5.1.3)of Λ∞2 C∞is called an elementary wedge.The action of a ∈gl∞, g ∈Gl∞is given as usual byρ(a) · (v0 ∧v−1 ∧v−2 ∧.

. . ) = (a · v0 ∧v−1 ∧v−2 ∧· · · +v0 ∧(a · v−1) ∧v−2 ∧· · · + .

. .ρ(g) · (v0 ∧v−1 ∧v−2 ∧.

. . ) = (g · v0) ∧(g · v−1) ∧(g · v−2) ∧.

. .

. (5.1.4)This does not extend to representations of the algebra gllf∞and the group Gllf∞,because of the infinities that occur (e.g., in the naive action of λ Pi∈Z Eii ∈Gllf∞on Λ∞2 C∞, for |λ| > 1).

However these infinities are relatively innocuous and after“renormalization” one obtains a projective representation ˆρ of gllf∞and Gllf∞, or,what is the same thing, a representation of a central extension of the algebra andgroup. In terms of the algebra generators it readsˆρ(Eij) = ρ(Eij) −δijθ0i,(5.1.5)where θ0i = 0 if i > 0 and 1 otherwise.

This defines a non trivial central extensionˆgllf∞= gllf∞⊕C of the Lie algebra gllf∞. The corresponding group will be describedin the next section.Let the fermion operators ψ(i), ψ∗(i) be the linear operators on Λ∞2 C∞that acton elementary wedges byψ(i) · (ǫi0 ∧ǫi1 ∧ǫi2 ∧.

. . ) = ǫi ∧ǫi0 ∧ǫi1 ∧ǫi2 ∧.

. .ψ∗(i) · (ǫi0 ∧ǫi1 ∧ǫi2 ∧.

. . ) =∞Xk=0(−1)kδiikǫi0 ∧ǫi1 ∧ǫi2 ∧· · · ∧ˆǫik ∧.

. .

(5.1.6)Now fix an integer n and a partition n. Then we have, as in section 3.1, a relabelingof the basis vectors and a corresponding relabeling of the fermion operators: thek-component fermions ψa(i), ψ∗a(i), i ∈Z, 1 ≤a ≤k correspond to ψ(j), ψ∗(j)whenever ǫa(i) corresponds to ǫj. The anti-commutation relations for the relabeledfermions are{ψb(i), ψc(ℓ)} = {ψ∗b(i), ψ∗c(ℓ)} = 0,{ψ (i) ψ∗(ℓ)}δ δ(5.1.7)

24M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992The action of Eℓmbc on semi-infinite wedge space is given by normal ordered fermionbilinears: if v ∈Λ∞2 C∞then ˆρ(Eℓmbc )v =: ψb(ℓ)ψ∗c(m) : v, where the normal orderingof fermions is defined by: ψb(ℓ)ψ∗c(m) :def=(ψb(ℓ)ψ∗c(m)if m > 0−ψ∗c(m)ψb(ℓ)if m ≤0 . (5.1.8)Recall the operators (Λ±a )j, j ̸= 0 of (3.2.1).

They generate, via the representationˆρ, on Λ∞2 C∞a Heisenberg algebra, the generators of which we will denote byαa(±j) := ˆρ((Λ±a )j) =Xℓ∈Z: ψa(ℓ)ψ(ℓ± j) :,j > 0, a = 1, . .

., k.(5.1.9)It is natural to introduce the operatorαa(0) :=Xℓ∈Z: ψa(ℓ)ψ∗a(ℓ) :,a = 1, . .

., k.(5.1.10)One checks that[αa(j), αb(ℓ)] = jδj+ℓ,0δab,j, ℓ∈Z, 1 ≤a, b ≤k. (5.1.11)Furthermore we need linear invertible operators ˆQa, 1 ≤a ≤k, on Λ∞2 C∞thatsatisfy the following defining relations:ˆQa · v0 = ψa(1) · v0,ˆQaψb(i) ˆQ−1a= ψb(i + δab),ˆQaψ∗b(i) ˆQ−1a= ψ∗b(i + δab).

(5.1.12)In (5.1.12) v0 is a distinguished element of Λ∞2 C∞, the 0th vacuum. In general onedefines the jth vacuum by:vj = ǫj ∧ǫj−1 ∧ǫj−2 ∧.

. .

. (5.1.13)We have, for ra the numbers (3.2.1) associated to j,αa(0) · vj = ravj,a = 1, .

. ., k.(5.1.14)The operators ˆQa are called fermionic translation operators.

They satisfy the fol-lowing relations ([tKvdL]):{ ˆQa, ˆQb} = 0,for a ̸= b(5.1.15)Introduce next fermionic fields, formal power series with operator coefficients:ψa(z) :=Xℓ∈Zψa(ℓ)zℓ,ψ∗a(z) :=Xψ∗a(ℓ)z−ℓ,a = 1, . .

., k.(5.1.16)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.25Now we can express the fermion fields completely in terms of the Heisenberg gen-erators αa(k) and the fermionic translation operators ˆQa. The parity operator onΛ∞2 C∞is defined byχ = (−1)Pb αb(0).

(5.1.17)The parity operator acts as (−1)k on the charge k sector Λ∞2k C∞of Λ∞2 C∞. Itfollows from Theorem 1.3 of [tKvdL] that we have the following bosonization for-mulae:ψa(z) = χ ˆQa(−z)(αa(0)+1) exp −Xℓ<01ℓz−ℓαa(ℓ)!exp −Xℓ>01ℓz−ℓαa(ℓ)!,ψ∗a(z) = ˆQ−1a χ(−z)−αa(0) exp Xℓ<01ℓz−ℓαa(ℓ)!exp Xℓ>01ℓz−ℓαa(ℓ)!,a = 1, .

. ., k.(5.1.18)So the only way the various bosonizations of k-component fermions are distin-guished is through the zero-modes αa(0), which, by (5.1.14), are able to detectwhich partition we are using.

One can use this “bosonized” form for the fermionsto express the whole representation of the Lie algebra gl∞in terms of the Heisen-berg operators αa(i) and the fermionic translation operators, [tKvdL]. We won’tneed this in the sequel.5.2 Central extension of Gllf∞and group action on Λ∞2 C∞.We have now defined an action on Λ∞2 C∞of a central extension of gllf∞, describedby an exact sequence0 →C →ˆgllf∞π∗→gllf∞→0.

(5.2.1)In this subsection we sketch the construction of the corresponding group and itsaction on Λ∞2 C∞, following the approach of [SeW, PrS], to which we refer for moredetails.The reason that the usual action of g ∈Gllf∞on v0∧v−1∧v−2∧. .

. by gv0∧gv−1∧gv−2∧.

. .

does not work is that the set {gvi}i≤0 is, in general, not admissible, even if{vi}i≤0 is. Now an admissible set {vi}i≤0 in C∞, if the vi are linearly independent,is the basis for the finite order part W fin for some (unique) W ∈Gr.

In the sameway {gvi}i≤0 is a basis for (gW)fin, (not necessarily admissible). The idea is now toreplace the possibly non admissible basis {g.vi} by an admissible one, say {wi}, andto replace the wedge v0∧v−1∧.

. .

by w0∧w−1∧. .

. .

Because of the ambiguity in thechoice of this admissible basis, we obtain in this way a projective representation ofGllf∞, or, equivalently, a representation of a central extension of this group. Belowwe will make this precise.Denote the group of invertible matrices of size Z≤0 × Z≤0 with a finite numberof nonzero lower triangular diagonals by:Gl(H0)lf = {a =Xi,j≤0aijEij | aij = 0 if i −j ≫0, a invertible}.

(5.2.2)A subgroup of Gl(H0)lf isT{t ∈Gl(H )lf | t1 + finr}(5 2 3)

26M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Here and in the sequel “finr” denotes a finite rank matrix. A finite rank matrix inGl(H0)lf is one with only a finite number of nonzero columns.Every W ∈Gr has an admissible basis for W fin, for instance the canonicalbasis of (2.2.4).

It will be convenient to think of an admissible set as a matrixv = (. .

. v−2v−1v0) of size Z × Zi≤0, with the vi as columns.

On such matrices wehave a right action of Gl(H0)lf. It is easy to see that if v and v′ are admissiblebases of W fin then v′ = v · t for t ∈T .

We will use frequently that v and v′ arebases for the finite order part W fin of the same W iffthe corresponding wedges vand v′ differ by a constant. Moreover if ˜v is any basis for W fin then we can find ana ∈Gl(H0)lf such that ˜v · a−1 is the canonical basis.

In particular if, for admissiblev, g · v = {gvi}i≤0 is not admissible, then we can find an a ∈Gl(H0)lf such thatg · v · a−1 is admissible.This leads to the introduction ofE = {(g, a) | g ∈Gl0,lf∞, a ∈Gl(H0)lf, g−−−a = finr}. (5.2.4)Here we write g ∈Gllf∞in block form with respect to the decomposition H =H0 ⊕H⊥0 :g =g−−g−+g+−g++.

(5.2.5)The subgroup of g ∈Gllf∞such that g−−: H0 →H0 is a Fredholm operator ofindex 0, is denoted by Gl0,lf∞and is the called the identity component of Gllf∞. Onechecks that (g1, a1) · (g2, a2) = (g1g2, a1a2) gives E a group structure.Suppose(g, a) ∈Gl0,lf∞× Gl(H0)lf and let v be an admissible basis corresponding to a wedgeof charge 0.

Then gva−1 is admissible iff(g, a) ∈E.The inclusion t ∈T 7→(1, t) ∈E gives us an exact sequence:1 →T →E →Gl0,lf∞→1. (5.2.6)Note that every t ∈T has a determinant.Lemma 5.2.1.

LetT1 = {t ∈T | det(t) = 1},E1 = {(1, t) ∈E | t ∈T1}. (5.2.7)Then T1 is normal in Gl(H0)lf and E1 is normal in E.Proof.

We must check that for all a ∈Gl(H0)lf ata−1 belongs to T1 if t does.Since t = 1 + f, f = finr we have ata−1 = 1 + afa−1 ∈T , so it remains tocheck that det(ata−1) = 1. Since a might not have a determinant we need a littleargument for this.

It runs as follows: write t = 1 + f as above and choose a basis{v1, v2, . .

.} of H0 such that for k > k0 all basis elements vk belong to the kernelof f and put V = ⊕ki=1vk.

Then V is a finite dimensional subspace of H0 and weget by restriction and projection a map t|V : V →V . One checks that det(t|V ) isindependent of the choices made here, so we can define det(t) = det(t|V ).

Definethen V ′ = a · V , t′ = ata−1, so that det(t′) = det(t′|V ′). Since a|V : V →V ′ is anisomorphism of finite dimensional vector spaces we see that det(t) = det(t′), as wewanted to prove.For the second part we must check, for all (g, a) in E and (1, t) in E1, that(g a) (1 t) (g−1 a−1)(1 ata−1) belongs to Ebut this follows from the first

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.27part.■Taking quotients we obtain from (5.2.6) the exact sequence:1 →T /T1 ≃C∗→E/E1 →Gl0,lf∞→1. (5.2.8)This gives us a central extension ˆGl0,lf∞:= E/E1 of the identity component of Gllf∞.To get a central extension of the whole group Gllf∞we need the shift automor-phism given byσ(g) = Λ · g · Λ−1.

(5.2.9)The semi-direct product Gl0,lf∞⋉σ Z (with multiplication (g, k)·(h, l) = (gσk(h), k +l)) is isomorphic to Gllf∞by the map (g, k) 7→g · Λk. The shift automorphism σlifts to an automorphism ˆσ of E/E1 as follows: an element of E/E1 can be writtenas (g, aT1), with g ∈Gl0,lf∞, a ∈GL(H0)lf, such that g−−−a = finr.

Then letˆσ((g, aT1)) := (σ(g), ¯σ(a)T1) where ¯σ(a) is obtained by adding to a a row andcolumn of zeroes and a diagonal 1:¯σ(a) =a001∈Gl(H0)lf. (5.2.10)One checks that this is independent of the representation of the coset aT1.Tosee that this indeed defines an automorphism note that the fiber of the projectionE/E1 →Gl0,lf∞over g is C∗and that ˆσ defines a homomorphism from the fiber overg to the fiber over σ(g) with kernel 1.

Therefore this homomorphism has to be anisomorphism and hence ˆσ is an automorphism.Next one defines ˆGllf∞:= ˆGl0,lf∞⋉ˆσ Z and we get an exact sequence1 →C∗→ˆGllf∞→Gllf∞→1,(5.2.11)and this is the central extension of Gllf∞we were looking for. One checks that thisexact sequence corresponds to the Lie algebra extension (5.2.1).Next we have to define an action of ˆGllf∞on Λ∞2 C∞.

It suffices to establish anaction on wedges v0 ∧v−1 ∧v−2 ∧. .

. , since as soon that is known we can extendto all of Λ∞2 C∞by linearity.

We first discuss wedges in some more detail.To a non zero wedge v = v0 ∧v−1 ∧v−2 ∧. .

. we can associate (non uniquely)a linear independent admissible set {vi}i≤0.

If v is a wedge of charge k then thecorresponding matrix v is of the formv = Λ−kv−v+,v−= 1 + finr,v+ = finr. (5.2.12)Here the subscripts ± refer to the decomposition of a Z × Z≤0 matrix into blocksinduced by the decomposition H = H0⊕H⊥0 , cf.

(5.2.5). Denote by A the collectionof all matrices of the form (5.2.12) with linearly independent columns.

On A wehave an action from the right of the group T1 of (5.2.7). We can identify, in abijective manner, a non trivial wedge v with an orbit of T1 in A via v ↔vT1.

Inorder to define an action of ˆGllf∞on Λ∞2 C∞it therefore suffices to define an actionon the orbit space A/T

28M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Let ˆQ ∈ˆGllf∞= ((e, eT1), 1) be the canonical lift of the shift matrix Λ−1 =P Ei+1i ∈Gllf∞.Just as any element of Gllf∞can be written as g · Λ−k, withg ∈Gl0,lf∞we can write an element of ˆGllf∞as (g, aT1) · ˆQk, (g, aT1) ∈ˆGl0,lf∞. Theaction of ˆQ reads in terms of elements of A/T1:ˆQj ·Λ−kv−v+T1= Λ−(k+j)v−v+T1.

(5.2.13)Next we define the action of (g′, a′T1) ∈ˆGl0,lf∞. We should have(g′, a′T1) ·Λ−kv−v+T1= ˆQk ·ˆQ−k(g′, a′T1) ˆQk·v−v+T1(5.2.14)Now note that ˆQ−k(g′, a′T1) ˆQk is an element of ˆGl0,lf∞, so it is of the form (g, aT1).To complete the definition of the ˆGllf∞action we put(g, aT1) ·v−v+T1 = g ·v−v+· a−1T1.

(5.2.15)We leave to the reader the easy verification that these definitions make sense, i.e.,are independent of the choice of representatives for the coset aT1 and for the orbitv−v+T1, and that the right hand side of (5.2.15) indeed belongs to A/T1.5.3 The projection π : ˆGllf∞→Gllf∞and the translation group.In the previous subsection we have constructed the central extension ˆGllf∞of Gllf∞.With the help of the fermions that act on Λ∞2 C∞we will describe the projectionπ : ˆGllf∞→Gllf∞very explicitly. Using this we show that the fermionic translationoperator ˆQa (5.1.10) that occurs in the bosonization formula (5.1.16) belongs tothe group ˆGllf∞and projects to the shift operator Qa of (3.2.4).

The fermionictranslation operators ˆQa are the ingredients for the lift of the translation groupT n ⊂Gllf∞to the central extension ˆGllf∞.Proposition 5.3.1. Let g = P gijEij ∈Gllf∞and let ˆg be any lift of g in ˆGllf∞.Then, for all i ∈Z, we have, as operators on Λ∞2 C∞,ˆg · ψ(i) · ˆg−1 =Xjgjiψ(j).Proof.

We write as before (g, aT1)· ˆQk for an element of ˆGllf∞, with ˆQ the canonicallift of the shift matrix Λ−1, see (5.2.13). It is clear thatˆQ · ψ(i) · ˆQ−1 = ψ(i + 1),i ∈Z.

(5.3.1)Hence the proposition holds for elements of ˆGllf∞of the form ˆg = ˆQk, k ∈Z. Itremains to prove the theorem for (g aT ) ∈ˆGl0,lf

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.29If we represent a wedge v ∈Λ∞2 C∞by an orbit vT1 then the action (5.1.6) ofψ(i) amounts to adding the vector ǫi on the right to v:ψ(i) · vT1 = (v | ǫi)T1. (5.3.2)This is independent of the choice of v: If t ∈T1 then (vt | ǫi)T1 = (v | ǫi)t001T1 =(v | ǫi)T1.

The proposition then states that the endomorphism ˆg · ψ(i) · ˆg−1 actson an orbit vT1 by adding on the right the column vector g · ǫi. This is a smallcalculation:(g, aT1) · ψ(i) · (g−1, a−1T1) · vT1 = (g, aT1) · ψ(i) · (g−1va)T1= (g, aT1) · (g−1va | ǫi)T1= ˆQ ·ˆQ−1(g, aT1) ˆQ· ˆQ−1 · (g−1va | ǫi)T1= ˆQ · (ΛgΛ−1, ¯σ(a)T1) · (Λg−1va | Λǫi)T1= ˆQ · (Λva | Λg · ǫi)¯σ(a)−1T1= ˆQ · (Λv | Λg · ǫi)T1= (v | g · ǫi)T1.

(5.3.3)■On Λ∞2 C∞there exists a unique positive definite Hermitian form ⟨. | .

⟩suchthat the elementary wedges (5.1.3) are orthonormal. With respect to this form theadjoint of ψ(i) is ψ∗(i) and the adjoint of ˆρ(Eij) is ˆρ(E†ij) = ˆρ(Eji).

To discuss theadjoint action of ˆGllf∞on ψ∗(i) we need a concrete description of this Hermitianform.Lemma 5.3.2. Let v, w be wedges in Λ∞2 C∞with charge k, l respectively.

LetvT1 and wT1 be the corresponding orbits in A/T1. Then⟨v | w⟩= δkl det(v†w)(5.3.4)Proof.

Let us define for the moment by H(v, w) = δkl det(v†w) a Hermitian formon Λ∞2 C∞. To show that H coincides with the standard Hermitian form we mustcheck that the elementary wedges are orthonormal for H. To this end let, for i ∈Zand λ ∈C, Ai(λ) = (exp(λEii), 1T1) ∈ˆGl0,lf∞.

Then we haveH(Ai(λ)v, w) = δkl det((Ai(λ)v)†w) = δkl det(v†Ai(λ∗)w) = H(v, Ai(λ∗)w)(5.3.5)For an elementary wedge v = ǫi0 ∧ǫi−1 ∧ǫi−2 ∧. .

. , with i0 > i−1 > i−2 > .

. .

, wehaveAi(λ) · v =(eλvif i ∈{i0, i−1, i−2, . .

. },vif i /∈{i0, i−1, i−2, .

. .

}. (5.3.6)Let w = ǫj0 ∧ǫj−1 ∧ǫj−2 ∧.

. .

, with j0 > j−1 > j−2 > . .

. , be another elementarywedge and let i = max(i0, j0).

If i0 > j0 we haveexp(λ)H(v w)H(A (λ∗)v w)H(v A (λ)w)H(v w)(5 3 7)

30M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992and hence H(v, w) = 0, whereas when i0 < j0 we haveexp(λ)H(v, w) = H(v, Ai(λ)w) = H(Ai(λ∗)v, w) = H(v, w)(5.3.8)and also in this case H(v, w) = 0. Repeating this argument for all other pairs(i−1, j−1), (i−2, j−2), .

. .

, we find that H(v, w) ̸= 0 implies v = w. Finally letv = ǫi0 ∧ǫi−1 ∧· · · ∧ǫi−N−1 ∧ǫ−N−k ∧ǫ−N−k−1 ∧. .

. be an elementary wedge ofcharge k, with the first −N exterior factors different from those of the kth vac-uum.

Then H(v, v) = det(M †M), where M is the Z × N matrix with columnsǫi−N−1, . .

. , ǫi−1, ǫi0.

It is clear that M †M = 1, proving that H makes the elemen-tary wedges orthonormal.■If g ∈Gllf∞then the necessary and sufficient condition for the Hermitian con-jugate matrix g† also to belong to Gllf∞is that g contains only a finite number ofnonzero upper triangular diagonals. If this condition is satisfied we say that g hasfinite width.

Suppose now that g has finite width and let ˆg = (g, aT1) be a lift ofg. Then a ∈Gl(H0)lf automatically also has finite width (and a† ∈Gl(H0)lf).

Wecalculate the adjoint of ˆg with respect to the Hermitian form of Λ∞2 C∞:⟨ˆgv | w⟩= det((gva−1)†w)= det((a−1)†v†g†w)= det(v†g†w(a−1)†)= ⟨v | ˆg†w⟩,(5.3.9)withˆg† := (g†, a†T1) ∈ˆGl0,lf∞. (5.3.10)So we see that the adjoint of ˆg with respect to the standard Hermitian form ofΛ∞2 C∞is a lift of the Hermitian conjugate matrix g†.Let now ˆg be the lift of a finite width element.

Then we have the followinganalogue of the Proposition 5.3.1ˆg · ψ∗(i) · ˆg−1 =Xj(g−1)ijψ∗(j). (5.3.11)Indeedˆg · ψ∗(i) · ˆg−1 = ((ˆg−1)† · ψ(i) · ˆg†)†= (Xj(g−1)†jiψ(j))†=Xj(g−1)ijψ∗(j)(5.3.12)We have derived here (5.3.10) under the assumption that g has finite width, theonly situation in which we will use this formula.

We leave it to the reader to provethis result for general g.Recall now the elements Qa of Gllf∞defined in (3.2.4). These act on the relabeledbasis introduced in section 3.1 byQǫ (j)ǫ (j + δ)a b1 2kj ∈Z(5 3 13)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.31Denote by ˆQ′a any lift of Qa to ˆGllf∞.Then by proposition 5.3.1 and equation(5.3.10) we findˆQ′aψb(i)( ˆQ′a)−1 = ψb(i + δab),ˆQ′aψ∗b (i)( ˆQ′a)−1 = ψ∗b(i + δab). (5.3.14)Comparing this with (5.1.10) we see that ˆQ′a has the same adjoint action on thefermions as the fermionic translation operators introduced in section 5.1, that hada priori no relation to ˆGllf∞.

To compare the action of ˆQ′a and ˆQa on Λ∞2 C∞weneed a lemma.Lemma 5.3.3. Let ˆQ′a be a lift of Qa.

ThenˆQ′a · v0 = νψa(1)v0,ν ∈C∗(5.3.15)Proof. The 0th vacuum v0 = ǫ0 ∧ǫ−1 ∧ǫ−2 ∧.

. .

corresponds to the orbit v0T1,where we can choose v0 = (. .

. ǫ−2ǫ−1ǫ0).

The columns for this matrix v0 form anadmissible basis forHfin0=Mj≤0Cǫj =kMb=1Mi≤0Cǫb(i). (5.3.16)Now if ˆQ′a is any lift of Qa then the wedge ˆQ′a·v0 corresponds to the orbit v′T1, withthe columns of v′ an admissible basis for (Qa · H0)fin.

Since Qa = Pb̸=a 1b + Λ−a ,and Λ−a ǫb(i) = ǫb(i + δab), we see that(Qa · H0)fin = Cǫa(1) ⊕kMb=1Mi≤0Cǫb(i). (5.3.17)Hence a particular simple admissible basis for (Qa · H0)fin is {ǫa(1)} ∪{ǫb(i) |b = 1, 2, .

. ., k; i ≤0}, corresponding to the orbit (v0 | ǫa(1))T1 and to the wedgeψa(1)v0.

Any two admissible bases of (Qa · H0)fin correspond to wedges that differby at most a non zero factor.■This lemma, combined with (5.3.13) and (5.1.10), shows that for any lift ˆQ′a ofQa we have ˆQa = ν ˆQ′a for some ν ∈C∗. HenceProposition 5.3.4.

Let ˆQa be the fermionic translation operators on Λ∞2 C∞de-fined in (5.1.10). Then ˆQa ∈ˆGllf∞and ˆQa projects to Qa ∈Gllf∞.Next we turn to the translation operators Tα responsible for the discrete timeevolution on the Grassmannian.

Since in the central extension the ˆQa no longercommute (as the Qa do) we have to make a choice here. We think of the rootlattice R = ⊕k−1i=1 Zαi as a sublattice of the rank k lattice ⊕ki=1Zδi through theidentification αi = δi −δi+1, i = 1, 2, .

. ., k. The δj are orthogonal for a symmetricbilinear form (·|·).

We can then write uniquely for any α ∈Rα =ℓXpijδij +kXnijδij,(5.3.18)

32M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992with the pij non negative and the nij strictly negative. We make then a choice ofthe ordering in the two sets of subscripts:i1 < i2 < .

. .

iℓ;iℓ+1 > · · · > ik. (5.3.19)Let ˆF = ⟨ˆQi, i = 1, 2, .

. ., k⟩be the group generated by the fermionic translationoperators.

We then define, using the ordering (5.3.19), a map R →ˆF, α 7→ˆTα byˆTα := ˆQpi1i1 ˆQpi2i2 . .

. ˆQpiℓiℓˆQniℓ+1iℓ+1 .

. .

ˆQnikik . (5.3.20)Clearly ˆTα is then a lift of the translation operator Tα (defined in subsection 3.2)and the group ˆT n generated by the ˆTαi is a central extension of the Abelian groupT n by Z2, defined by a cocycle ǫ : R × R →Z2 given byˆTα ˆTβ = ǫ(α, β) ˆTα+β(5.3.21)So ǫ satisfies the cocycle properties ǫ(α, β)ǫ(α + β, γ) = ǫ(α, β + γ)ǫ(β, γ) andǫ(α, 0) = ǫ(0, α) = 1.

The choice (5.3.19) was made to ensure simple properties ofthe cocycle ǫ, as described in the following Lemma.Lemma 5.3.5. The cocycle ǫ satisfies for all α, β ∈R:(1) ǫ(α, −α) = 1,(2) ǫ(α, β)ǫ(β, α) = (−1)(α|β),(3) ǫ(−α, −β) = (−1)(α|β)ǫ(α, β).The proof of the Lemma consists of simple computations using the fact that theˆQi satisfy the anti-commutation relations (5.1.15) and the fact that the sum of thecoefficients pij, nij is zero.

Note that part (1) of the Lemma implies the usefulrelation ˆT −1α= ˆT−α for all α ∈R.5.4 Group decomposition and τ-functions.In this section we will study the Gauss decomposition in Gllf∞by means of theaction of the central extensionˆGllf∞on Λ∞2 C∞.In fact we will need a slightrefinement: if g ∈Gllf∞with Gauss decomposition g = g−Pg+ then we can write(uniquely) g+ = h · gs+, with h a diagonal matrix and gs+ a upper triangular matrixwith ones on the diagonal to get a decompositiong = g−Phgs+. (5.4.1)Note that in the Gauss decomposition the factors g−, g+ are in general not uniquelydefined.

However, the element h here is uniquely determined; later on we will seethat its entries are essentially the τ-functions of W = gHj (when W is in the Hjcell), to be defined in Definition 5.4.3 below. For any lift ˆg of g we can find elementsˆg−, ˆP, ˆh and ˆgs+ projecting to the corresponding elements without a hat to getˆg = ˆg−ˆP ˆhˆgs+.

(5.4.2)This decomposition is however not unique: each of the factors can be multipliedby a non zero complex constant as long as the product of these factors is one. Thefollowing lemma fixes a normalization

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.33Lemma 5.4.1. Let g = g−Phgs+ be a fixed factorization (5.4.1).Assume thatPHj = Hj.

Then, for any lift ˆg of g there is a unique factorization as in (5.4.2)such that(1) ˆgs+vj = vj. (2) ˆPvj = vj.

(3) ⟨vj | ˆg−vj⟩= 1Proof. For the first part note that for any lift ˆgs+ of gs+ the wedge ˆgs+vj correspondsto an admissible basis for gs+Hj = Hj, since gs+ is upper triangular.

As before weuse that two wedges corresponding to two admissible bases of the same point of Grdiffer by a non zero factor, so that ˆgs+vj = νvj. By changing the lift we can makeν = 1.

The proof for the second part is the same. For the third part we note thatg−has finite width, and g†−is an upper triangular element of Gllf∞that by (1) hasa unique lift ˆg†−∈ˆGllf∞such that ˆg†−vj = vj.

Then the adjoint of ˆg†−is the uniquelift of g−that satisfies (3).■To decide whether or not PHj = Hj one can use the following Lemma.Lemma 5.4.2. Let ˆP be any lift of a permutation matrix P. Then⟨vj | ˆPvj⟩̸= 0 ⇐⇒PHj = HjProof.

The wedge ˆPvj corresponds to some admissible basis for PHj. Since P isa permutation matrix it is clear that PHfinjhas a basis consisting of basis vectors{ǫσP (i)}i≤j, where σP is the permutation associated to P. Let SjP = {σP (i) | i ≤j}and order the elements of SjP as i0 > i−1 > i−2 > .

. .

. Then {vim}m≤0 is anadmissible basis of PHfinjand we haveˆPvj = νǫi0 ∧ǫi−1 ∧ǫi−2 ∧.

. .

,ν ∈C∗(5.4.3)Since elementary wedges are orthogonal we have⟨vj | ˆPvj⟩̸= 0 ⇐⇒ˆPvj = νvj,ν ∈C∗⇐⇒σP restricts to a permutation of {i | i ≤j}⇐⇒PHj = Hj(5.4.4)■Consider next the following lift toˆGllf∞of wn0 (t, α):ˆwn0 (t, α) = exp(Xi>0kXa=1tiaαa(i)) · ˆTα ∈ˆGllf∞. (5.4.5)Let ˆg ∈ˆGllf∞be any lift of g ∈Gllf∞.

Then we defineˆg(t, α) = ˆwn0 (t, α)−1ˆg. (5.4.6)

34M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Definition 5.4.3. Fix an integer j, a positive integer n, a partition n of n and letW = gHj ∈Gr.

Fix a lift ˆg of g. The τ-function of type j, n of W is the functionon Γn given as “vacuum expectation value”:τ j,nW (t, α) := ⟨vj | ˆg(t, α) · vj⟩. (5.4.7)Note that if ˆg′ is another element ofˆGllf∞projecting to g the τ-function calculatedwith ˆg′ will differ from (5.4.7) by a constant.

This will be irrelevant in the sequel.Also note that the τ-function is defined for all values of (t, α), also when W(t, α)does not belong to the big cell.In fact the τ-function of type j, n determineswhether or not W(t, α) belongs to the Hj cell:Proposition 5.4.4. Let j, n, W be as above.

Let Γn be the group of evolutionsof type n on Gr and let Γj,nW be the subset of (t, α) ∈Γn such that pr : W(t, α) =g(t, α)Hj →Hj is an isomorphism. (see section 2.2).

Then for all (t, α) ∈Γn thefollowing two statements are equivalent:(1) τ j,nW (t, α) ̸= 0,(2) (t, α) ∈Γj,nW .Proof. We use the Gauss decomposition of ˆg(t, α) ∈ˆGllf∞.

We suppress for sim-plicity the reference to (t, α) and write ˆg = ˆg−ˆP ˆhˆgs+, with the normalization as inLemma 5.4.1. Then, writing τ for τ j,nW (t, α):τ = ⟨vj | ˆg−· ˆP · ˆh · ˆgs+ · vj⟩= ⟨(ˆg−)† · vj | ˆP · ˆh · vj⟩= exp(λj)⟨vj | ˆP · vj⟩.

(5.4.8)Here we have used Lemma 5.4.1, and the fact that vj is an eigenvector of elementsof lifts of diagonal matrices: ˆh · vj = exp(λj)vj, for some λj (depending on (t, α)).Using Lemma 5.4.2 we see from (5.4.7) that τ j,nW (t, α) ̸= 0 iffPHj = Hj. ButPHj = Hj iffW(t, α) = g−(t, α)Hj.Now W(t, α) = g−(t, α)Hj, for g−(t, α)strictly lower triangular, is equivalent to W(t, α) belonging to the Hj cell.■Note that from the proof of this proposition it follows that for (t, α) ∈Γj,nW wehave, if ˆg(t, α) = ˆg−· ˆP ˆh · ˆgs+,ˆPˆhgs+ · vj = τ j,nW (t, α)vj,(5.4.9)a fact that will be used in the calculation of the wave function in terms of τ-functions.5.5 Relation between wave function and τ-function.

In the construction of thewave function of a point W of the Grassmannian the lower triangular part gj−ofthe Gauss decomposition adapted to Hj occurs (see definition 4.4.1). It is a rathertrivial observation that one can calculate matrix elements of gj−by lifting it to thecentral extension Gllf∞and using the semi-infinite wedge space (see the introductionfor the finite dimensional situation).

First note that there is a unique lift of gj−suchthat ⟨vj | ˆgj−vj⟩= 1. The proof is as in Lemma 5.4.1.

To find the coefficient ofthe matrix Epq in gj−we observe that Epq is represented on Λ∞2 C∞by the operatorψ ψ∗(if p ̸q) The following Lemma is then maybe not too surprising

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.35Lemma 5.5.1. Fix an integer j and a positive integer n and a partition n of n.Let g ∈Gllf∞and let g(t, α) be given by (4.1.4).

Assume that g(t, α) belongs tothe Hj cell, see section 2.2. Let gj−(t, α) ∈Gllf∞be the minus component of theGauss decomposition of g(t, α) of Lemma 2.2.2 adapted to j.

Write gj−(t, α) =P gpq(t, α)Epq.Let ˆg(t, α) be an arbitrary lift of g(t, α) and lift gj−(t, α) to theunique element ˆgj−(t, α) ∈ˆGllf∞that satisfies ⟨vj | ˆgj−(t, α)vj⟩= 1. Then for allp ∈Z, q ≤j we havegpq(t, α) = ⟨(ψpψ∗q) · vj | ˆgj−(t, α) · vj⟩,(5.5.1.a)= ⟨(ψpψ∗q) · vj | ˆg(t, α) · vj⟩/τ j,nW (t, α),(5.5.1.b)where τ j,nW (t, α) is the τ-function of type (j, n) of W = gHj.Proof.

For simplicity we will mostly suppress the dependence on (t, α). If p ̸= q,and p ≤j, then, by definition of gj−, we have gpq = 0, but also (ψpψ∗q) · vj = 0, soin that case (5.5.1a) is true.

In case p = q ≤j we have gpp = 1, (ψpψ∗p) · vj = vjand (5.5.1a) holds because of the normalization of the lift ˆgj−.Assume next that p > j. Then we use Lemma 5.3.1 to calculate the right handside of (5.5.1a).

The elementary wedge (ψpψ∗q) · vj = ǫj ∧ǫj−1 ∧· · · ∧ǫq+1 ∧ǫp ∧ǫq−1 ∧. .

. corresponds to the orbit vp,qj T1, where vp,qjis the Z × Z≤0 matrix withcolumns (.

. .

ǫq−1ǫpǫq+1 . .

. ǫj).

The wedge ˆgj−vj corresponds to an orbit vT1, wherewe can take v to bev = gj−Λ−j10a−1,(5.5.2)for a the identity matrix. With this choice v is a Z × Z≤0 matrix of the form(.

. .

v−2v−1v0) with for all i ≤0vi = gj−· ǫj+i. (5.5.3)Therefore⟨(ψpψ∗q) · vj | ˆgj−· vj⟩= det(vp,qj )†v),= detgpqgpq−1.

. .gpj01.

. .0............00. .

.1= gpq,(5.5.4)proving (5.5.1.a) also in this case.Now gj−(t, α) is a factor in the decompositiong(t, α) = gj−· f j · P · g+ = gj−· f j · P · h · gs+,(5.5.5)where f j is defined in (2.2.4) and we have decomposed g+ in a diagonal part h andan upper triangular part gs+ with ones on the diagonal. Since gj−· f j is the factorg−in the Gauss decomposition and (t, α) ∈Γj,nW we can apply Lemma 5.4.1 to getfor any lift ˆg(t, α) a unique factorizationˆg(t α)ˆgjˆf jˆPˆh ˆgs(5 5 6)

36M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992such thatˆPvj = vj,ˆgs+vj = vj,⟨vj | ˆgj−· ˆf jvj⟩= 1. (5.5.7)Now f jHj = Hj so we have ˆf jvj = νvj, with ν ∈C∗.But because of thenormalization of ˆgj−we must have in fact ν = 1 and ˆf jvj = vj.

This implies thatˆgj−· vj = ˆg(t, α) · (ˆgs+)−1 · ˆh−1 · ˆP −1 · ( ˆf j)−1 · vj= ˆg(t, α) · vj/τ j,nW (t, α),(5.5.8)using (5.4.8).■Note that only because we have assumed that gj−is the minus part of the Gaussdecomposition adapted to j we are able to calculate its matrix elements gpq in asimple way in the semi-infinite wedge representation: if we had used the ordinaryGauss decomposition of (2.2.2) the finite matrix in (5.5.4) would not have hadzeroes below the diagonal.In Lemma 5.5.1 we calculated the coefficients of gj−in an expansion in termsof the standard basis Epq. After the relabeling of type n, so that gj−= P gbcrsErsbc ,we find of course for the new coefficients an expression in terms of the relabeledfermions: if ǫc(s) corresponds to ǫq, with q ≤j, thengbcrs = ⟨(ψb(r)ψ∗c(s)) · vj | ˆg(t, α) · vj⟩/τ j,nW (t, α),(5.5.9)for all 1 ≤b ≤k, r ∈Z.

In particular the columns of type c of gj−that occur inthe definition of the wave function of type j, n have s = rc so that the coefficientsthat occur in the wave function are of the form gbcrrc for c = 1, 2, . .

., k. This will beused in the next Theorem,Theorem 5.5.2. Fix g ∈Gllf∞, a lift of g to ˆg ∈ˆGllf∞, an integer j, a positive inte-ger n and a partition n of n into k parts and let W = gHj ∈Gr.

Let wj,nW (z; t, α),τ j,nW (t, α) be the wave function and the τ-function associated to these data. Then,if we write wj,nW (z; t, α) = Pkb,c=1 wbcEbc, we have for all (t, α) ∈Γj,nW :wbc = ⟨ψ∗c(rc)vj | ˆwn0 (t, α)−1ψ∗b(z)ˆg⟩/τ j,nW (t, α).

(5.5.10)More explicitly we havewbc(z; t, α) = (−1)rb−rcz−(α|δb)z−rb(−z)δbc−1ǫ(α, β) exp(Xℓ>0zℓtbℓ)τ j,nW (t⟨b⟩, α + β)τ j,nW (t, α),(5.5.11)whereτ j,nW (t⟨b⟩, α + β) = exp(−Xℓ>0z−ℓℓ∂∂tbℓ)τ j,nW (t, α + β),(5.5.12)where β = δb −δc ∈R is the root corresponding to the root vector Ebc ∈sl(k, C),and ǫ is the cocycle defined in (5.3.21).Proof. For simplicity we suppress the reference to (t, α) and mostly also to z in theproof.

Now by definition of the wave function we have for the cth columnwbc = etb · w0(z; t, α) · n(gj−· ǫc(rc)),= exp(Xtbizi)z<α,δb> Xgbcrrcz−reb. (5.5.13)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.37Hence by (5.5.9) we haveXr∈Zgbcrrcz−r =Xr∈Zz−r⟨ψb(r)ψ∗c(rc) · vj | ˆg(t, α) · vj⟩/τ j,nW (t, α),= ⟨Xr∈Zzrψb(r)ψ∗c(rc) · vj | ˆg(t, α) · vj⟩/τ j,nW (t, α),= ⟨ψb(z)ψ∗c(rc) · vj | ˆg(t, α) · vj⟩/τ j,nW (t, α),= ⟨ψ∗c(rc) · vj | ψ∗b(z)ˆg(t, α) · vj⟩/τ j,nW (t, α). (5.5.14)Here we have implicitly extended the Hermitian form ⟨· | ·⟩on Λ∞2 C∞to a map⟨· | ·⟩1 : Λ∞2 C∞[[z, z−1]] × Λ∞2 C∞→C[[z, z−1]] such that ⟨zu | v⟩1 = z−1⟨u | v⟩.There is a similar map ⟨· | ·⟩2 : Λ∞2 C∞× Λ∞2 C∞[[z, z−1]] →C[[z, z−1]] such that⟨u | zv⟩2 = z⟨u | v⟩and we have⟨ψb(z)u | v⟩1 = ⟨u | ψ∗b(z)v⟩2.

(5.5.15)In (5.5.14) we have dropped the subscripts 1, 2, as we will continue to do in thesequel. Now, ˆg(t, α) = ˆwn0 (t, α)−1ˆg andψ∗b(z) ˆwn0 (t, α)−1 = exp(Xi>0−tbizi)z⟨α,δb⟩ˆwn0 (t, α)−1ψ∗b(z),(5.5.16)as follows from (see (5.1.12,5.3.20)),ˆTαψ∗b(z) ˆT −1α= z⟨α,δb⟩ψ∗b(z),[αa(i), ψ∗b(z)] = −ziδabψ∗b (z).

(5.5.17)Using this, and the explicit form (4.3.5-6) for w0(z), in (5.5.14) gives part (5.5.10)of the theorem.We next continue with the calculation of w′bc(z) := ⟨ψ∗c(rc)·vj | ψ∗b(z)ˆg(t, α)·vj⟩.The fermion operator ψ∗c(rc) is the coefficient of z−rc in the field ψ∗(z), so using(5.1.14), (5.1.17) and the bosonization formula (5.1.18) we getψ∗c(rc) · vj = ˆQ−1c (−1)j−rc · vj(5.5.18)This gives, also using the fact that the fermionic translation operators ˆQc are uni-tary,w′bc(z) = (−1)j−rc⟨vj | ˆQcψ∗b(z)ˆg(t, α) · vj⟩. (5.5.19)In the fermion field ψ∗b(z) the operator (−z)−αb(0) occurs.

We need to move this tothe left to let it act on vj. We have in general[αa(0), ˆQb] = δab ˆQb,(5.5.20)soˆQ ˆQ−1( z)−αb(0)ˆQ ( z)−(αb(0)+1) ˆQ−1( z)−(αb(0)+1)+δbc ˆQ ˆQ−1(5 5 21)

38M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992This givesw′bc(z) = (−1)j−rc⟨(−z)(αb(0)+1)−δbcvj | ˆQc ˆQ−1b χ exp Xℓ<01ℓz−ℓαb(ℓ)!·,· exp Xℓ>01ℓz−ℓαb(ℓ)!ˆg(t, α) · vj⟩,= (−1)−rc(−z)δbc−rb−1⟨exp Xℓ>0−1ℓz−ℓαb(ℓ)!vj | ·· ˆQc ˆQ−1b ˆg(t⟨b⟩, α) · vj⟩,= (z)−rb(−1)−rb−rc(−z)δbc−1⟨vj | ˆQc ˆQ−1b ˆg(t⟨b⟩, α) · vj⟩,(5.5.22)where t⟨b⟩is as in (5.5.12) and we have used that αb(ℓ)vj = 0 if ℓ> 0.Finally we have by (5.3.20) and Lemma 5.3.5:ˆQc ˆQ−1bˆT −1α= ˆT −1βˆT −1α ,= ˆT−β ˆT−α,= ǫ(−β, −α) ˆT−(α+β),= ǫ(α, β) ˆT −1α+β. (5.5.23)Putting this all together gives then the rest of the theorem.■Note that in approach of, say, [DJKM] the formula (5.5.10) would be the defini-tion of the wave function, while in the approach of [Di3] the τ-function would bedefined by a formula like (5.5.11).The relation between the wave function and the τ-function given by Theorem5.5.2 allows us also to express the coefficients of resolvents, lattice resolvents, etc.in terms of τ-functions.

For instance for the example of the Davey-Stewartson-Todasystem discussed in section 4.6 we find that the first coefficient wm1 of the expansion(4.6.6) of the wave operator is given bywm1 =−∂11 log(τ m)−τ m−1/τ m−τ m+1/τ m−∂12 log(τ m). (5.5.24)(For simplicity we use the j = 0 part of the Grassmannian and write τ m(t) forτ 0,nW (t, mα1), see section 4.6, and suppress the time dependence).

From (4.6.9–10)we then see thatqm = −2τ m−1/τ m,rm = 2τ m+1/τ m.(5.5.25)A small calculation using the diagonal part of δm, see (4.6.9), then proves that theexpression of the variable Qm in terms of τ-functions is given byQm = ∂2 log(τ m). (5.5.26)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.396.Multicomponent KP equations in bilinear form and Pl¨ucker equa-tions.6.0 Introduction. In the previous section we have found a relation between thewave function of a point of the (polynomial) Grassmannian and τ-functions, matrixelements of the fundamental representations of gl∞.

These wave functions are so-lutions of the linear equations (4.4.6). In this section we will reformulate the linearequations (4.4.6) in terms of bilinear equations of Hirota type: any (formal) solu-tion of the linear equations is at the same time a solution to the bilinear equations(Proposition 6.1.1).

This allows us to make the connection with the Pl¨ucker equa-tions of the embedding of Gr in PΛ∞2 C∞, which are equations for the τ-function.So we get, just as in the one component case, three equivalent descriptions of themulti component KP hierarchy: the wave functions of the multi component KPhierarchy satisfy both the linear equations and the bilinear equations and they canbe expressed in terms of τ-functions that satisfy there own, equivalent, system ofequations.6.1Dual Grassmannian and Pl¨ucker equations. The equations (4.4.6) for thewave function wW (t, α) can be formulated in terms of a bilinear equation involvingalso the so called dual wave function w∗W (t, α).

This bilinear equation amountsto an orthogonality relation between W ∈Gr and W ∗, an element of the dualGrassmannian Gr∗, as was explained in [HP] in an analytic context for the KP-hierarchy. So we start out this subsection by briefly discussing the dualization of allour constructions.

Then we derive, for completeness sake, the bilinear equations,show that the bilinear equations for the wave function and its dual are equivalentto the differential difference linear equations (4.4.6) and give the connection to thePl¨ucker equations of the embedding Grj →PΛ∞2j C∞.The starting point of the theory developed until now was the space H of infinitecolumn vectors, see (2.1.2). The dual notion isH∗= {∞Xi=mciǫ∗i | ci ∈C, m ∈Z},(6.1.1)where ǫ∗i is the linear function on H given by ⟨ǫ∗i , ǫj⟩= δij, i, j ∈Z.

We can think ofH∗as consisting of infinite row vectors and the natural bilinear pairing H∗×H →Cis then just matrix multiplication between a row and column vector.On H∗we have the action of g ∈Gllf∞given byg · ǫ∗i = ǫ∗i g−1 =∞Xk=mǫ∗k(g−1)ik. (6.1.2)Define, cf.

(2.2.1),H∗j = {∞Xk=j+1ckǫ∗k} ⊂H∗. (6.1.3)Then Gr∗is the collection of W ∗⊂H∗of the form g · H∗j = H∗j g−1 for someg ∈Gllf∞, j ∈Z.

There is a natural 1-1 correspondence between points of Gr andGr∗: W = gHj corresponds to W ∗= g · H∗j . One checks that such W and W ∗areorthogonal with respect to the pairing ⟨⟩since H and H∗are

40M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992We define also, given a partition n, a relabeling on H∗: put ǫ∗j = ǫ∗a(i) if ǫj = ǫa(i)in H. Then we have a map to k-component row vector Laurent series, cf. (4.3.1):∗n : H∗→H∗(k),n(ǫ∗a(i)) = zi−1e∗a,(6.1.4)where e∗a, a = 1, 2, .

. ., k is a basis for the k-component row vectors and H∗(k) =⊕C((z))e∗a.

The bilinear pairing between H∗and H then translates into the residuepairing on H∗(k) × H(k):(f ∗, g) = Resz(f ∗(z)g(z)),f ∗∈H∗(k), g ∈H(k),(6.1.5)where Resz is the coefficient of z−1 in a formal power series.On the dual Grassmannian we have the time evolution of type n given by W ∗7→W ∗(t, α) := wn0 (t, α)−1 · W ∗= W ∗wn0 (t, α), with wn0 (t, α) given by (4.1.1). Anelement W of Gr belongs to the Hj cell iffthe corresponding element W ∗of thedual Grassmannian belongs to the H∗j cell.

In particular the set Γj,nW of elements(t, α) of the evolution group such that W(t, α) belongs to the Hj cell is equal toΓj,nW ∗.The wave function wW of type j, n was defined using the columns gj−· ǫb(rb(j)),with rb(j) given by (3.2.1). The analogous dual numbers are r∗b(j) = rb(j) + 1,with the following interpretation: ǫ∗b(rb) is the basis vector of type b occurring inH∗j with smallest argument.

Note that∗n(ǫ∗b(r∗b)) = zr∗b −1e∗b = zrbe∗b. (6.1.6)Now we define the dual wave function using the rows of (gj−)−1 and the numbersr∗b: for (t, α) ∈Γj,nWw∗W (t, α) := ∗n (ǫ∗1(r∗1)ǫ∗2(∗r2) .

. .ǫ∗k(r∗k))t(gj−)−1· w0(t, α)−1(6.1.7)The dual wave function can be written in terms of the dual wave operator, actingnow from the left (cf.

(4.5.1)):w∗W (t, α) = −→w W (t, α)·w0(t, α)−1,−→w W = (1k×k+Xi>0w∗i−→∂−i)diag(zr1, zr2, . .

., zrk). (6.1.8)Denote by W ∗(k) the image of W ∗in H∗(k).

Then the rows of w∗W (t, α) belong toW ∗(k) for all (t, α) ∈Γj,nW , just as the columns of wW (t, α) belong to W (k). BecauseW ∗(k) and W (k) are orthogonal for the residue pairing we find the bilinear identity:Resz(w∗W (t, α)wW(t′, α′)) = 0,(t, α), (t′, α′) ∈Γj,nW .

(6.1.9)Both the linear equations (4.5.6) for the wave function and the bilinear equations(6.1.9) are derived in apparently different ways from the geometry of the Grassman-nian. In fact these equations are equivalent.

Indeed, consider two matrix PDO’sacting from the left and right:−→P =Xpi∂i,←−Q =X ←−∂iqi,pi, qi ∈Matn(C). (6.1.10)Then we define the dual operators to act in the opposite direction:(−→P )∗:=Xpi(−←−∂)i,(←−Q )∗:=X(−∂)iqi.

(6.1.11)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.41Proposition 6.1.1. Letw(z; t, α) = w0(z; t, α) · ←−w ,←−w = diag(z−r1, .

. ., z−rk) · (1k×k +Xi>0←−∂−iwi),w∗(z; t, α) = −→w · w0(z; t, α)−1,−→w = (1k×k +Xi>0w∗i−→∂−i) · diag(zr1, .

. ., zrk),with wi, and w∗i matrices of size k. If w, w∗are both defined and infinitely differ-entiable for (t, α) and (t′, α + γ), then the bilinear equationsResz(w∗(z; t, α)w(z; t′, α + γ)) = 0(6.1.12)are equivalent to the equations−→w = (←−w −1)∗,(6.1.13a)∂ibw = w · (Rib)+,(6.1.13b)w(z; t, α + γ) = w(z; t, α) · (Uγ)−1+ ,(6.1.13c)where Rib = ←−w −1 · ←−Λ ib · ←−w , Uγ = ←−w −1 · ←−T −1γ· ←−w .We omit the proof, which is rather similar to the one for the KP hierarchy, cf.

[Di2].In Theorem 5.5.2 we found an expression for the wave function in terms of matrixelements of the semi-infinite wedge space. There is, as one might expect, a similarexpression for the dual wave function in terms of some dual semi-infinite wedgespace.

However the situation is simpler than that. Note that w∗W consists essentiallyof rows of g−1−, and since g−= 1+X, with X ∈glj∞−, we have g−1−= 1−X, becauseX2 = 0.

Hence we can calculate the matrix elements of g−1−and thus w∗W in termsof the standard semi-infinite wedge space. Writing w∗W = Pkbc w∗bcEbc we find byessentially the same calculation as in Theorem 5.5.2 thatw∗bc = z−1⟨ψ∗b(r∗b)vj | ˆwn0 (t, α)−1ψc(z)ˆgvj⟩/τ j,nW (t, α).

(6.1.21)Consider the orbit Oj of the group ˆGl0,lf∞through the vacuum vj in Λ∞2 C∞.The projectivization of Oj can be identified with the component Grj of Gr. It iswell known ([KP]) that the points of Oj are characterized as follows: τj ∈Λ∞2j C∞belongs to Oj iffXi∈Zψ(i)τj ⊗ψ∗(i)τj = 0.

(6.1.22)So the equation (6.1.22) might be called the Pl¨ucker equation for the embedding ofGrj in PΛ∞2j C∞. Note that if we use the bosonization formulae (5.1.18) we obtaindifferential-difference equations for the τ-function.

We refrain from writing downexplicit equations.Using the relabeled fermionic fields of (5.1.16) we can write this asResz[z−1kXψc(z)τj ⊗ψ∗c(z)τj] = 0. (6.1.23)

42M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Multiplying this by ˆwn0 (t, α) ⊗ˆwn0 (t′, α′)/τ j,nW (t, α)τ j,nW (t′, α′) and taking the innerproduct with the element ⟨ψb(r∗b)vj | ⊗⟨ψ∗d(rd)vj | givesResz[kXc=1⟨ψb(r∗b)vj | ˆwn0 (t, α)ψc(z)τjτ j,nW (t, α)⟨ψ∗d(rd)vj | ˆwn0 (t′, α′)ψ∗c(z)τjτ j,nW (t′, α′)] = 0, (6.1.24)which is, using (5.5.10, 6.1.21), the bilinear identity (6.1.9). In other words thePl¨ucker equation for the embedding of Grj in PΛ∞2j C∞is nothing but the bilinearequation for a point W of the component Grj.7.

Reduction to loop groups and KdV type equations.7.0 Introduction. In the previous sections we have found for every W in theGrassmannian and for every choice of partition of n into k parts a solution of thedifferential-difference k-component KP hierarchy.In the classical 1-component KP case one can for every integer n impose con-straints on solutions of the KP to obtain solutions of the n-KdV hierarchy, relatedto ˜Gllf(n, C), the loop group of Gl(n, C).

This procedure is called n-reduction andamounts to considering ˜Gllf(n, C) in a natural way as a subgroup of Gllf∞. This givesrise to a subspace Grn of Gr, called the n-periodic Grassmannian.

The embeddingof ˜Gllf(n, C) in Gllf∞and Grn are discussed in section 7.1.In the general case that we are studying we can similarly impose for every par-tition n of n into k parts constraints on the k-component KP to obtain solutions ofwhat one might call the n-reduced KP-hierarchy. This is discussed in section 7.2.In section 7.3 the n-reduced KP hierarchy is rewritten in terms of n×n matricesdepending on a variable λ, so as to make the relation with the loop group ˜Gllf(n, C)and the n-periodic Grassmannian explicit.

The n-reduced KP hierarchy rewritten inthis way will be called the n-KdV hierarchy. Every conjugacy class in the symmetricgroup is determined by a partition n and determines a gradation of ˜gllf(n, C).

In[Wi2] Wilson proposes to construct for every gradation of ˜gllf(n, C) a hierarchyof differential equations. We show that our n-KdV hierarchies are essentially theequations Wilson had in mind for the gradation corresponding to n. (He was dealingwith the modified equations, related to the infinite n-periodic flag manifold in asimilar way as our n-KdV hierarchies are related to the n-periodic Grassmannian.Also he didn’t discuss the discrete part of the hierarchies.

)7.1 n-periodicity. Recall from section 4.4 the map n from the space H of infinitecolumn vectors to H(k), the space of k-component formal Laurent series in thevariable z, constructed using the partition n. Here we will consider the space H(n)of n-component formal Laurent series in the variable λ: let ˜ei, 1 ≤i ≤n be a basisfor Cn, defineH(n) := ⊕ni=1C((λ))˜ei,(7.1.1)and introduce the isomorphism H →H(n) byǫj 7→λ−p−1˜eq,(7.1.2)where j = np+q, 1 ≤q ≤n.

This corresponds to the construction of H(k) of section4.4 for the partition of n into n parts. Let ˜gllf(n, C) be the collection of formalloops of the form P∞λiA with A ∈gl(n C) and m some integer This forms in

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.43the natural way a Lie algebra and we can define a (Lie) algebra homomorphism˜gllf(n, C) →gllf∞byλp ˜Eij 7→Xℓ∈ZEn(ℓ−p)+i,nℓ+j,(7.1.3)where ˜Eij is the element of the natural basis of n×n matrices with as only nonzeroentry a 1 on the ijth place. (In general we will use a tilde ˜ to indicate that anobject is related to the loop algebra, loop group or anything “of size n”).The image of this homomorphism consists of the n-periodic elements, i.e., thoseX in gllf∞such thatX = ΛnXΛ−n,(7.1.4)where Λ = P Eii+1 is the shift matrix in Gllf∞.

If we expand X as X = P XijEijthen the condition (7.1.4) implies for the coefficients Xi+n,j+n = Xij for all i, j ∈Z.In the same way we get a surjective homomorphism from the formal loop group˜Gllf(n, C) (consisting of the invertible elements of ˜gllf(n, C)) to the subset of n-periodic elements of Gllf∞.Since H(k) and H(n) are both isomorphic to H there is an isomorphism n,k :H(k) →H(n) given explicitly byeaz−i 7→λ−p−1˜et,(7.1.5)where i = nap + q, 1 ≤q ≤na and t = n1 + · · · + na−1 + q. Using this isomorphismwe can translate, of course, the action of k × k matrices on H(k) into an action onH(n).

In particular the diagonal matrix zEaa acting on H(k) corresponds to theaction of the n × n block diagonal matrixPa = diag(0n1, . .

., 0na−1, Pna, 0na+1, . .

., 0nk). (7.1.6)where 0nc is the zero matrix of size nc × nc and Pna is the na × na matrixPna =010.

. .00001. .

.00..................000. . .10000. .

.01λ00. .

.00. (7.1.7)Since we have seen in subsection 4.4 that the generator Λ+a of the pre-Heisenbergalgebra Hn of type n corresponds in H(k) to zEaa we find that the action of thepositive part of the pre-Heisenberg algebra in H(n) is generated by the elementsPa.

The action of the pre-translation group is similarly generated by the elements˜Tαi = P−1ni + Pni+1 + Pj̸=i,i+1 1nj, 1 ≤i ≤k.We have seen that the shift matrix Λ does not correspond under n to an elementof the formal loop group of size k, leading to the introduction of the numbers rbof (4.3.1). Now, when we use the map (7.1.2), the situation is much simpler: theelement Pn of ˜Gl(n, C) is the image of the shift matrix Λ.

This means in particularthat, if we denote by H(n)jthe image of Hj in H(n), we have, cf. (4.3.3),H(n) = P −jH(n) = P −j ⊕nC[[λ]]˜e(7 1 8)

44M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Let g ∈Gllf∞be n-periodic and let W = gHj. Since ΛnHj = Hj−n ⊂Hj we haveΛnW = ΛngHj = gΛnHj ⊂W.

Conversely, elements of Gr satisfying ΛnW ⊂Wcan be obtained from Hj by an n-periodic group element (see [PrS]) and are calledalso n-periodic. The collection of n-periodic elements in Gr is denoted by Grn, andGrn is called the n-periodic Grassmannian.7.2 The n reduced k-component KP hierarchy.The infinite shift matrix Λn ∈Gllf∞corresponds to multiplication by diag(zn1, .

. .

, znk)in H(k) and to multiplication by λ = λ1n×n in H(n). For simplicity we also writeoften λ for diag(zn1, .

. ., znk) in H(k).

So if W ∈Grn we have for the image W (k) inH(k) the relation λW (k) ⊂W (k). In particular for the wave function, the columnsof which belong to W (k) when (t, α) ∈Γj,nW , we haveλwW (t, α) ⊂W (k),(7.2.1)where we say that a matrix belongs to W (k) if its columns do.

Now by Proposition4.4.2 this means that there is for every positive integer ℓa unique k × k matrixdifferential operator M (ℓ), such thatλℓwW (t, α) = wW · M (ℓ)(t, α),(7.2.2)Just as wW also M (ℓ) is defined for (t, α) ∈Γj,nW .One sees that the resolvent (4.5.5) satisfieswW · Ra = zEaawW ,(7.2.3)so we have, using that λ = diag(zn1, . .

., znk),wW · (M (ℓ) −kXa=1Raℓna) = 0. (7.2.4)By the proof of Proposition 4.4.2 this means thatM (ℓ) =kXa=1Raℓna,ℓ> 0.

(7.2.5)So in the linear combination of resolvents on the right hand side of (7.2.5) thenegative powers of ←−∂that might occur cancel to give a differential operator.Introduce now the derivation∂ℓ,n =kXa=1∂taℓna,ℓ> 0. (7.2.6)Then we have∂ℓ,nwW = wW ·kXa=1Rℓnaa+,wM (ℓ)(7.2.7)

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.45On the other hand, writing wW = w0 · ←−w W , with ←−w W the wave operator, we have∂ℓ,nwW = w0 ·kXa=1zℓnaEaa· ←−w W + w0 · ∂ℓ,n←−w W ,= wW · M (ℓ) + w0 · ∂ℓ,n←−w W . (7.2.8)Comparing (7.2.7) and (7.2.8) we find that the wave operator ←−w W and hence allresolvents Ra, lattice resolvents Uαi ((4.7.5)) and the multi-component KP operatorL of (4.5.11) are independent of the variables corresponding to ∂ℓ,n, ℓ> 0 (in the n-periodic case).

Conversely if W ∈Gr produces a solution of the multi-componentKP hierarchy that is independent of these variables it belongs to the n-periodicGrassmannian.Definition 7.2.1. The n-reduced k-component differential-difference KP hierarchyis the system of deformation equations for the k × k matrix PDO L of the formL = A←−∂+ O(←−∂0) given by∂ibL = [L, (Rib)+],L(t, α + αi) = (Uαi)+ · L(t, α) · (Uαi)−1+ .

(7.2.9a)together with the condition∂ℓ,nL = 0,ℓ> 0. (7.2.9b)So we get for every n-periodic element W ∈Grn a solution LW = w−1W A←−∂wWof the n-reduced k-component KP hierarchy.7.3 The n-KdV hierarchy.In this section we want to give an alternative description of the n-reduced k-component KP hierarchy.In the case k = 1 there are (at least) two other de-scriptions available of the n-reduced hierarchies: the scalar Lax equation approach,involving nth order scalar differential operators, and the approach using first ordermatrix differential operators (see, e.g., [DS]).

We discuss both methods in the gen-eral case of an arbitrary partition. It will turn out that the scalar Lax operatormethod does not generalize in a satisfactory way.The k × k matrix differential operator M (ℓ) introduced in (7.2.2) satisfies thesame equations as the pseudo-differential operator L (4.5.11):∂ai M (ℓ) = [M (ℓ), (Rai)+],M (ℓ)(t, α + αi) = (Uαi)+ · M (ℓ)(t, α) · (Uαi)−1+ .

(7.3.1)If we consider the partition of n into 1 part (the principal partition), then we havefor M := M (1) the relation M = Ln and L is the nth root of the operator M. (Wetake A = 1 here). As is well known in this case the equations (7.3.1) form thegeneralized n-KdV hierarchy (the discrete part is now, of course, absent) and thishierarchy is equivalent to the n-reduced 1-component KP hierarchy, cf., [SeW].

Sothe matrix M seems to be the natural generalization of the Lax operator in theprincipal case.However, in general the equations (7.3.1) are just a consequence of the equations(7 2 9) and not equivalent to them since the operator M contains less information

46M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992than L. In fact in the extreme case where n is the partition of n into n parts(the homogeneous case) M is just ←−∂1n×n. This is so because then the differentialoperator M is the sum of the pseudo differential operators Ra that are of the form←−∂Eaa + [Eaa, w1] + O(←−∂−1).

So the scalar Lax operator formulation of n-reducedKP does not seem to generalize simply to the general case, cf., [Di1]. Thereforewe now sketch how the n-reduced k-component KP hierarchy (7.2.9) fits in theframework of n × n matrix differential operators.Recall the construction of the k-component KP hierarchy: we started with a Win the Grassmannian, mapped it to W (k) in H(k) and considered the natural flowW (k) 7→w0(t, α)−1 · W (k).

Then we considered the subspace W (k)fin of elements ofW (k) of the form w0(t, α) times a finite order (in z) k-component vector. The spaceW (k)fin was stable under the action of ←−∂and we proved in Proposition 4.4.2 thatW (k)fin was in fact a free rank k module over C[←−∂], with basis the columns of thewave function wW .

This lead in Proposition 4.5.2 to linear equations for the wavefunction and this in turn produced the k-component KP hierarchy.Now, in the n-periodic case, W (k)fin is not only invariant under the action of ←−∂but also under the action of λ = diag(zn1, zn2, . .

., znk), i.e., W (k)fin is a C[λ] module.It turns out that W (k)fin is free of rank n and we can give an explicit basis. Thenthe time evolution of this basis will as before lead to linear equations and we willobtain in very much the same way as before a collection of differential differenceequations, the n-KdV hierarchy.

In the k-component KP case the objects one dealswith are k × k matrices over ←−∂(since W (k)fin is rank k over C[←−∂]) whereas in then-KdV case we deal with n × n matrices over λ (since now we think of W (k)fin as arank n module over C[λ]).Now we turn to some of the details of the construction. Fix an integer j andconsider an n-periodic element g of Gllf∞and the corresponding element W = gHjof Grn.

We embed W in H(n) using the map (7.1.2), so that we now deal withn-component vectors depending on λ.We say that an element ˜g ∈˜Gllf(n, C) is in the ˜Hj cell if˜g = ˜gj−· ˜gj+,(7.3.4)where˜gj−= P −jn ˜1n×n +Xi<0λi˜gi!P jn,˜gj+ = P −jnXi≥0λi˜giP jn,˜gi ∈gl(n, C), ˜g0 ∈Gl(n, C). (7.3.5)This is called the j-Birkhoffdecomposition of ˜g.

An element ˜g is in the ˜Hjcelliffthe corresponding (under the map (7.1.3)) n-periodic element of Gllf∞is in theHj cell.The image of ˜gj−(resp.˜gj+) in Gllf∞does not quite coincide with theminus component gj−(resp. gj+) of the Gauss decomposition of g adapted to Hj,since gj−(gj+) is not n-periodic, in general.

So in case g is n-periodic we have twonatural decompositions adapted to Hj. However the columns j −n + 1, j −n +2j of the n periodic elements corresponding to ˜gj and ˜gj are equal to the

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.47same columns of the original factors gj−and gj+ of the Gauss decomposition of gadapted to Hj (the other columns will differ, in general). Since this are the onlycolumns that play a rˆole it is irrelevant which decomposition of g we use.

Besidesthe Birkhoffdecomposition in the loop group we will need in the sequel also thefollowing corresponding decomposition of the formal loop algebra:˜gl(n, C) = ˜gl(n, C)j−⊕˜gl(n, C)j+,(7.3.6)with (cf. (2.2.3) and (7.1.8)):˜gl(n, C)j−= P −jngl(n, C) ⊗λ−1C[λ−1]P jn,˜gl(n, C)j+ = P −jn(gl(n, C) ⊗C[[λ]]) P jn.

(7.3.7)Fix also a partition n of n and consider the time flow from the correspondingHeisenberg algebra Hn on H(n)generated by˜wn0 (t, α) = exp(Xi>0kXa=1tai Pia) ˜Tα,(7.3.8)The n × n loop group wave function of type j, n, associated to W and defined for(t, α) ∈Γj,nW , reads:˜wW (t, α) = ˜wn0 (t, α) · ˜gj−(t, α),(7.3.9)where ˜gj−is the minus component in the Birkhoffdecomposition for ˜Hj of ˜g(t, α) =˜wn0 (t, α)−1˜g.We can also describe ˜gj−by noting that the ith column ˜gj−· ˜ei isthe unique element of W (n)(t, α) of the form P −jn (˜eni + O(λ−1)).So the loopgroup wave function consists essentially of n columns of g−(t, α), whereas the wavefunction contains only k columns.The point of the introduction of the loop group wave function is that its columnsform a basis for ˜W (k)fin , the image of W (k)fin in H(n).Proposition 7.3.2. Let W be n-periodic.

Fix (t, α) ∈Γj,nW . Then W (k)fin is a freerank n module over the ring C[λ], with basis the columns of ˜wW (t, α).The proof of this Proposition is very much the same as that of Proposition 4.3.2and is omitted.Next we define n × n loop resolvents and lattice resolvents by˜Ra := ˜wW (t, α)−1 · Pa · ˜wW (t, α)˜Uαi := ˜wW (t, α)−1 · ˜T −1αi · ˜wW (t, α)(7.3.10)So the resolvent ˜Ra belongs to the loop algebra and the lattice resolvent ˜Uαi tothe loop group.

We use the convention that subscripts on ˜Ra will refer to the Liealgebra decomposition 7.3.7 for j = 0, while subscripts on ˜Uαi, when it is in the H0cell, refer to the Birkhoffdecomposition of type 0.Then the analogue of Proposition 4 4 2 is

48M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992Lemma 7.3.3. Let W ∈Grj and suppose that (t, α), (t, α + αi) belong to Γj,nW .Then the loop lattice resolvent ˜Uαi is in the H0 cell and we have:∂tbi ˜wW (t, α) = ˜wW (t, α) · ( ˜Rib)+˜wW (t, α + αi) = ˜wW (t, α) · ( ˜Uαi)−1+(7.3.11)The proof is the same as that of Proposition 4.4.2.Note that the derivations with respect to the times (7.2.6) act trivially on the˜wW , i.e., ∂ℓ,n ˜wW = ˜wW · P Pℓnaa= λℓ˜wW .The analogue of the operator LW that solves the k-component KP hierarchy isthe resolvent ˜L = ˜w−1w · P AaPa · ˜wW .

We could now define the n-reduced KdVhierarchy in terms of deformation equations for ˜L. However in our situation it turnsout that all information of ˜L is already contained in the in the positive part ˜L+and it is convenient to formulate the theory in terms of this.To proceed we need a little digression on finite order automorphisms and twistedloop algebras associated to a partition n. (For more details and proofs see [tKvdL]).We associate to every partition n a diagonal matrix in gl(n, C):Hn =kXa=1Ha,Ha =12nanaXj=1(na −2j + 1)Ejjaa.

(7.3.12)We use Hn to define an automorphism of gl(n, C):σn = exp(2πiad(Hn)) : gl(n, C) →gl(n, C). (7.3.13)If N ′ is the least common multiple of the parts of the partition n then the order Nof σn is equal to 2N ′ in case N ′( 1na +1nb ) is odd for some pair of parts na, nb andN is equal to N ′ otherwise.

Then σn defines a Z/NZ grading of gl(n, C):gl(n, C) = ⊕N−1i=0 gi,gi = {x ∈gl(n, C) | σn(x) = ωix},ω = exp(2πi/N). (7.3.14)Next we define the twisted loop algebra:L(g, σn) := {∞Xi=mµiy¯ı | y¯ı ∈g¯ı}.

(7.3.15)Write µ = exp(iθ/N) and put ΦHn = exp(−iθad(Hn)). Then ΦHn : L(g, σn) →˜gl(n, C) is an isomorphism between the twisted loop algebra L(g, σn) and the un-twisted loop algebra ˜gl(n, C), with λ = µN = exp(iθ) as loop variable.Define a Cartan subalgebra hn of gl(n, C) associated to n with basis Eia, a =1, 2, .

. ., k and i = 1, 2, .

. ., na, where Ea = Ena1aa+ Pna−1j=1 Ejj+1aa.

Under σn theCartan subalgebra hn is mapped to itself and so the decomposition (7.3.14) inducesa decomposition of hn. We can then define as in (7.3.15) the twisted loop algebraL(hn, σn).

The image of L(hn, σn) under the isomorphism ΦHn is precisely theHeisenberg algebra ˜Hn generated by Pa and Qa := P−1a . In particular the elementµN/naE corresponds to PFor different n one obtains distinct Heisenberg algebras

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.49in ˜gl(n, C) and one proves also that allHeisenberg algebras of ˜gl(n, C) (up toisomorphism) are obtained in this way.The decomposition gl(n, C) = hn ⊕h⊥n , where h⊥n is the direct sum of the rootspaces with respect to hn, induces a decompositionL(g, σn) = L(hn, σn) ⊕L(h⊥n , σn). (7.3.16)We also need the twisted loop group L(G, σn), the collection of units in L(g, σn).The image of L(G, σn) under the isomorphism ΦHn is the untwisted loop group˜Gl(n, C).

If ˜g ∈˜gl(n, C), or ˜g ∈˜Gl(n, C), then we write ˜gσn for the inverse imagein the twisted loop algebra or twisted loop group.Lemma 7.3.4. (1) If ˜g−= 1n + O(λ−1) then ˜gσn−= 1n + O(µ−1).

(2) Let ˜g−as above. Then there exist two formal loops uσn = Pi<0 µiui, vσn =Pi<0 µivi, with ui ∈(hn)¯ı, vi ∈(h⊥n )¯ı, such that˜gσn−= exp(uσn) exp(vσn).Proof.

The element Hn induces a Z grading of ˜gl(n, C) as follows: if y ∈g ¯m, then[Hn, y] = sy,s = ¯mN + ℓ,(7.3.17)for some ℓ∈Z, ¯m = 0, 1, . .

., N −1. We put thendeg(λry) = (r + s)N = (r + ℓ)N + ¯m.

(7.3.18)This is the grading that corresponds to the grading by powers of µ in L(g, σn) underthe standard isomorphism: we have: Φ−1Hn(λry) = µ(r+ℓ)N+ ¯my. One calculates thatthe eigenvalues of Hn are of the formqnb−pna+12na−12nb,(7.3.19)where 1 ≤p ≤na and 1 ≤q ≤nb.

From this one sees that the absolute valueof the eigenvalues of Hn is strictly less than 1, and that the only possibilities in(7.3.18) are m = 0, l = 0 or 0 < m ≤N −1 and l = 0, −1. This means thata homogeneous element of the form λ−iy, i > 0, has strictly negative degree andmaps to an element of L(g, σn) containing a strictly negative power of µ. Thisproves part (1).

The proof of part (2) is similar to that of Lemma 5.1 in [BtK].■We return to the formulation of the n reduced KP hierarchy in terms of n × nmatrix operators. Let ∂y = P Aa∂ta1 and Py = P AaPa.

Then one considers theoperatorDy(t, α) = ←−∂y −˜wW (t, α)−1∂y( ˜wW (t, α)) = ←−∂y −˜L(t, α)+,←−∂Pq(t α)(7.3.20)

50M.J. BERGVELT, A.P.E.

TEN KROODE,Urbana, December 14th, 1992for some matrix q(t, α). The components of q(t, α) are then considered the fun-damental fields of the theory.

Note that Py + q is the polynomial part (in λ) ofan element in the adjoint orbit through Py of the group of elements of the formg = 1+O(λ−1), and hence q is constant in λ. In particular the degree (7.3.18) of thehomogeneous components of q is strictly less than the maximum of Nna , a = 1, .

. ., k.The most general form for q is obtained if one chooses the constants Aa such thatPy = P AaPa is regular, i.e., such that the centralizer of it is just ˜Hn.

Howeverthe theory would work also if Py is not regular: then some of the components of qwould be zero.Lemma 7.3.5. Let Dy(t, α) as above and denote by Dσnythe inverse image of thisoperator in the twisted loop algebra L(g, σn): Dσny= ←−∂y−P AaµNna Ea−qσn.

Thenthere exists a unique formal power series vσn = Pi<0 µivi, with vi ∈(h⊥n )¯ı, suchthatexp(ad(vσn))(Dσny ) = ←−∂y −XAaµNna Ea +Xi

First note that trying to construct vσn directly from (7.3.21) by expandingin powers of µ seems not to lead in the usual way (cf. [DS]) to a simple recursionscheme for the vi because of the inhomogeneity of Py.

Therefore we break up Dyin pieces corresponding to the homogeneous components of Py.Indeed, let Py + q(t, α) = [Ad(˜g−)(Py)]+ for ˜g−= 1 + O(λ−1). Define Pa +qa = [Ad(˜g−)(Pa)]+ and Da = ←−∂ta1 −Pa −qa.

We have Dy = Pka=1 AaDa, andq = Pka=1 Aaqa. We will write as in Lemma 7.3.4 ˜gσn−= exp(uσn) exp(vσn).

Define˜w = ˜w0˜g−, so that ˜wσn = ˜w0 exp(uσn) exp(vσn) and we have˜wσnDσna= 0. (7.3.22)From this one easily sees that for all a:exp(ad(vσn))(Dσna ) = ←−∂ta1 −µNna Ea +Xi

[BtK]. Now fix an a between 1 andk.

Comparing the coefficient of µN/na−j on both sides of (7.3.23) we obtain anequation of the form[v−j, Ea] + k(a)N/na−j = polynomial in qσnaand inv−l, −l > −j, and ta1 derivatives. (7.3.24)Using the partition n we decompose v−j in blocks: write v−j = Pka,b=1 v−j,ab,where the matrix v−j,ab is zero outside the block with index ab of size na × nb.Then the commutator in the left hand side of (7.3.24) becomes Pkb=1 Aa[v−j,baCna−Cv] where CE+P Eis a cyclic matrix of size p Now C acts from

PARTITIONS, VERTEX OPERATOR CONSTRUCTIONS, Urbana, December 14th, 1992 . .

.51the left (right) as an invertible linear transformation on the space of p × q (q × p)matrices for any q. Furthermore the adjoint action of Cp is an invertible lineartransformation on the orthogonal complement of the linear span of the powers inCp in the space of p × p matrices.

Using these facts one can express the blocksv−j,ab, v−j,ba for b = 1, . .

., k and k(a)N/na−j uniquely in terms of qσnaand the v−l,−l > −j, and their ta1 derivatives. Letting now also a run from 1 to k we expressall blocks of v−j in terms of these variables.

Note that to find v−j,ab according tothis method one can consider the diagonalization (7.3.23) of Da or of Db. Thesemust give the same result, since the consistency of the procedure it ensured by theexistence of at least one solution vσn.

Induction leads now to the conclusion thatvσn and P kiµi (with ki = P Aak(a)i) are polynomials in qσnaand ta1 derivatives,a = 1, . .

., k. Next we use the fact (which follows from (7.3.22)) that0 = [Dy, Da] = [←−∂y −Py −q, ←−∂ta1 −Pa −qa],(7.3.25)This gives us an expression for ∂ta1(q) in terms of derivatives of qa with respect to y.Since the projection q 7→qa is just differentiation with respect Aa we find that wecan express all ta1 derivatives of qa in terms of y derivatives and the Lemma follows.■From the proof of the last Lemma it follows that all resolvents, being of the formR(p) = ˜w−1 · p · ˜w = ˜g−1−· p · ˜g−= exp(−ad(v))(p), are differential polynomials ofthe fundamental field q(t, α). This makes the following definition reasonable:Definition 7.3.6.

The n-KdV hierarchy is the collection of deformation equationsof the operator Dy (of the form (7.3.20) with Py +q(t, α) = [Ad(˜g−)(Py)]+ for some˜g−= 1 + O(λ−1)) given by∂tbi Dy = [( ˜Rib)+, Dy],Dy(t, α + αi) = ( ˜Uαi)+(Dy(t, α))( ˜Uαi)−1+ . (7.3.26)Here (Rib)+ and ( ˜Uαi)+ are the positive parts of the resolvents (7.3.10), using ˜w =˜w0˜g−instead of ˜wW .Note that we have made here the simple choice ∂y = P Aa∂ta1 to determine the“spatial variable” y in our system of equations (7.3.6).

It will be clear from theconstruction that we can make much more general choices for this spatial variableand still get a reasonable set of equations, cf. [FNR].As an example of a n-KdV hierarchy consider the 2-reduction of the Davey-Stewartson-Toda system of section 4.6, for the partition 2 = 1 + 1.

The resultingsystem is the AKNS-Toda system: the field Qm, being the second x derivativeof the τ-function, see (5.5.32), is in the 2-periodic case identically zero, since theτ-function is now independent of x.Furthermore, in the resolvents and latticeresolvents the operator ←−∂can be effectively replaced by the variable z; this isbecausewW←−∂= w0 · ←−w W←−∂= zw0 · ←−w W + w0∂(←−w W ) = wW ,(7.3.27)since also the wave operator is x independent in the 2-periodic case. This gives thetheory described in [BtK]

52M.J. BERGVELT, A.P.E.

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(Bergvelt) Department of Mathematics, University of Illinois, Urbana, IL 61820,USAE-mail address: bergv@huygens.math.uiuc.edu(ten Kroode) Nieuwe Prinsengracht 771, 1018 VR Amsterdam, The Netherlands

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