The Finite Non-periodic Toda Lattice: A Geometric and Topological Viewpoint

THE FINITE NON-PERIODIC TOD A LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 Abstract. In 1967, Japanese physicist Morik azu T oda published the seminal pap ers [78] and [79], exhibiting soliton solutions to a c hain of particles with nonlinear in teractions b etween nearest neighbors. In the decades that follow ed, T o da’s system of particles has been generalized in diﬀeren t directions, eac h with its o wn analytic, geometric, and topological c haracteristics that sets it apart from the others. These are known collectiv ely as the T o da lattice. This survey describes and compares several versions of the ﬁnite non-p erio dic T o da lattice from the persp ective of their geometry and topology . Contents 1. Outline of the pap er 2 2. Finite non-p erio dic real T o da lattice 3 2.1. Symmetric form 4 2.2. Hessen b erg form 6 2.3. Extended real tridiagonal symmetric form 7 2.4. Extended real tridiagonal Hessen b erg form 9 2.5. F ull Symmetric real T o da lattice 15 3. Complex T o da lattices 16 3.1. The momen t map 16 3.2. Complex tridiagonal Hessen b erg form 17 3.3. The full Kostan t-T o da lattice 21 3.4. Nongeneric ﬂo ws in the full Kostant-T oda lattice 26 4. Other Extensions of the T o da Lattice 28 4.1. Isosp ectral deformation of a general matrix 28 4.2. Gradien t formulation of T oda ﬂows 31 5. Connections with the KP equation 32 5.1. The τ -functions for the symmetric T o da lattice hierarch y 32 5.2. The KP equation and the τ -function 34 5.3. Grassmannian Gr ( k , n ) 37 6. The T o da lattice and integral cohomology of real ﬂag manifolds 39 6.1. The momen t p olytop e and W eyl group action 40 6.2. In tegral cohomology of G/B 42 6.3. Blo w-ups of the indeﬁnite T o da lattice on G and the cohomology of G/B 45 References 48 1 Partially supp orted by NSF gran t DMS0404931. 2 YUJI K OD AMA 1 AND BARBARA SHIPMAN 2 1. Outline of the p aper W e organize the pap er as follows: Section 2 introduces the real non-perio dic T o da lattices. It begins with t w o form ulations of the T o da lattice in whic h the ﬂow obeys a Lax equation on a set of real tridiagonal matrices whose sub diagonal en tries are positive. The matrices are symmetric in one formulation and Hessenberg in the other. The ﬂows exist for all time and preserv e the sp ectrum of the initial condition, and the system is completely integrable. The references cited for thes e results are [4, 27, 28, 38, 55, 56, 58, 76]. When the matrices of the T oda lattice system are extended to allow the sub diagonal en tries to tak e on any real v alue, the tw o forms of the T o da lattice diﬀer in the b ehavior of their solutions and in the top ology of their isosp ectral sets. In the symmetric case, the ﬂo ws exist for all time and the isosp ectral manifolds are compact, while in the Hessenberg form, the ﬂows blow up in ﬁnite time and the isosp ectral manifolds are not compact. F or these results, the cited references are [15, 17, 50, 51, 62, 63, 80]. Section 1 concludes with the full symmetric real T o da lattice, in whic h the ﬂo ws evolv e on the set of all real symmetric matrices. The symplectic structure comes from the Lie-P oisson structure on the dual of a Borel subalgebra of sl ( n, R ). In the full T o da lattice, more constan ts of motion are needed to establish complete in tegrabilit y . Cited references for this material are [22, 47, 77]. Section 3 describ es the iso-level sets of the constan ts of motion in complex T o da lattices. The complex tridiagonal T o da lattice in Hessen berg form diﬀers from the real case in that the ﬂows no longer preserve ”signs” of the subdiagonal entries. The ﬂows again blow up in ﬁnite (complex) time, and the isosp ectral manifolds are not compact. These manifolds are compactiﬁed b y embedding them into a ﬂag manifold in tw o diﬀerent w ays. In one compactiﬁcation of isosp ectral sets with distinct eigenv alues , the ﬂows enter low er-dimensional Bruhat cells at the blow-up times, where the singularity at a blow-up time is characterized by the Bruhat cell [25, 53]. Under a diﬀerent em b edding, whic h w orks for arbitrary spectrum, the T oda ﬂows generate a group action on the ﬂag manifold, where the group dep ends on how eigenv alues are repeated. The group is a pro duct of a diagonal torus and a unipotent group, b ecoming the maximal diagonal torus when eigen v alues are distinct and a unip otent group when all eigenv alues coincide [70]. These group actions, together with the momen t map [5, 34, 44], are used in [72] to study the geometry of arbitrary isospectral sets of the complex tridiagonal T oda lattice in Hessenberg form. These compactiﬁed iso-lev el sets are generalizations of toric v arieties [59]. F urther properties of the actions of these torus and unipotent comp onen ts of these groups are found in [30, 69, 71, 73, 74, 75]. The surv ey then considers the full Kostan t-T o da lattice, where the complex matrices in Hessen berg form are extended to ha v e arbitrary complex en tries b elow the diagonal. The techniques for ﬁnding additonal constan ts of motion in the real symmetric case are adapted to the full Kostant-T oda lattice to obtain a complete family of in tegrals in inv olution on the generic symplectic leav es; the geometry of a generic iso-level set is then explained using ﬂag manifolds [26, 33, 40, 67]. Nongeneric ﬂo ws of the full Kostan t-T o da lattice are described in terms of special faces and splittings of moment polytop es, and monodromy around nongeneric iso-lev el sets in sp ecial cases is determined [65, 66, 68, 70]. Section 4 provides other extensions of the T o da lattice. In this paper, w e only discuss those related to ﬁnite non-perio dic T o da lattices. W e ﬁrst in tro duce the ﬂow in the Lax form on an arbitrary diagonalizable matrix, which can be integrated b y the inv erse scattering method (or equiv alen tly b y the factorization metho d) [52]. W e then sho w how the tridiagonal Hessen b erg and symmetric T o da lattices, which are deﬁned on the Lie algebra of type A (that is, sl ( n )), are extended to semisimple Lie algebras using the Lie algebra splittings given by the Gauss decomposition and the QR decomp osition, resp ectively [12, 15, 16, 54, 36, 60]. As an imp ortant related hierarc hy of ﬂo ws, w e explain the Kac-v an Moerb eke system, which can b e considered as a square ro ot of the T oda lattice [35, 42]. W e also sho w that the Pfaﬀ lattice, which evolv es on symplectic matrices, is related to THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 3 the indeﬁnite T o da lattice [1, 48, 49]. As another asp ect of the T oda lattice, we men tion its gradient structure [7, 8, 9, 10, 11, 23], whic h explains the sorting property in the asymptotic behavior of the solution and can b e used to solv e problems in com binatorian optimization and linear programming [13, 37]. Section 5 explains the connections with the KP equation [41, 43] in the sense that the τ -functions as the solutions of the T o da lattices also pro vide a certain class of solutions of the KP equation in a W ronskian determinant form [6, 21, 32, 39, 46, 47, 57]. This is based on the Sato theory of the KP hierarc hy , which states that the solution of the KP hierarch y is given b y an orbit of the universal Grassmannian [64]. The main proposition in this section sho ws that there is a bijection b etw een the set of τ -functions that arise as a k × k W ronskian determian t and the Grassmannian Gr ( k , n ), the set of all k -dimensional subspaces of R n . W e present a method to obtain soliton solutions of the KP equation and giv e an elemen tary in tro duction to the Sato theory in a ﬁnite-dimensional setting. The momen t polytop es of the fundamen tal represen tations of S L ( n, C ) pla y a crucial role in describing the geometry of the soliton solutions of the KP equation. Section 6 sho ws that the singular structure giv en by the blow-ups in the solutions of the T o da lattice contains information ab out the integral cohomology of real ﬂag v arieties [18, 19]. W e b egin with a detailed study of the solutions of the indeﬁnite T o da lattice hierarch y [51]. The singular structure is determined b y the set of zeros of the τ -functions of the T o da lattice. First w e note that the image of the moment map of the isospectral v ariety is a conv ex p olytop e whose v ertices are the orbit of the W eyl group action [16, 31]. Each vertex of the polytop e corresp onds to a ﬁxed point of the T oda lattice, and it represents a unique cell of the ﬂag v ariety . Each edge of the polytop e can be considered as an orbit of the sl (2) T oda lattice (the smallest nontrivial lattice), and it represen ts a simple reﬂection of the W eyl group. There are tw o types of orbits, either regular (without blow-ups) or singular (with blow-ups). Then one can deﬁne a graph, where the v ertices are the ﬁxed p oints of the T o da lattice and where tw o ﬁxed p oints are connected by an edge when the sl (2) ﬂow b et ween them is regular. If the ﬂo w is singular, then there is no edge b et ween the tw o p oin ts. The graph deﬁned in this w a y turns out to b e the incidence graph that gives the in tegral cohomology of the real ﬂag v ariety , where the incidence n um b ers associated to the edges are either 2 or − 2 [45, 20]. W e also show that the total num b er of blow-ups in the ﬂow of the T o da lattice is related to the p olynomial asso ciated with the rational cohomology of a certain compact subgroup [14, 18, 19]. 2. Finite non-periodic real Toda la ttice Consider n particles, eac h with mass 1, arranged along a line at positions q 1 , ..., q n . Betw een eac h pair of adjacent particles, there is a force whose magnitude dep ends exp onentially on the distance b et ween them. Letting p k denote the momen tum of the k th particle, and noting that d dt q k = p k since eac h mass is 1, the total energy of the system is the Hamiltonian (2.1) H ( p, q ) = 1 2 n P k =1 p 2 k + n − 1 P k =1 e − ( q k +1 − q k ) . The equations of motion (2.2) dq k dt = ∂ H ∂ p k , dp k dt = − ∂ H ∂ q k , 4 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 yield the system of equations for the ﬁnite non-p erio dic T o da lattice, (2.3) dq k dt = p k , k = 1 , ..., n, dp k dt = − e − ( q k +1 − q k ) + e − ( q k − q k − 1 ) , k = 1 , ..., n. Here w e set e − ( q 1 − q 0 ) = 0 and e − ( q n +1 − q n ) = 0 with the formal b oundary conditions q 0 = −∞ , q n +1 = ∞ . In the 1970’s, the complete in tegrability of the T oda lattice w as disco v ered b y Henon [38] and Flasc hk a [27] in the context of the p erio dic form of the lattice, where the b oundary condition is tak en as q 0 = q n . Henon [38] found analytical expressions for the constan ts of motion following indications by computer studies at the time that the T oda lattice should be completely integrable. That same year, Flasc hk a [27, 28] (indep endently also b y Manak ov [56]) show ed that the p erio dic T o da lattice equations can b e written in Lax form through an appropriate change of v ariables. The complete in tegrabilit y of the ﬁnite non-perio dic T oda lattice was established b y Moser [58] in 1980. A system in Lax form [55] giv es the constants of motion as eigen v alues of a linear operator. In the ﬁnite non-p erio dic case, there are tw o standard Lax forms of the T oda equations. 2.1. Symmetric form. Consider the change of v ariables (Flasc hk a [27], Moser [58]) (2.4) a k = 1 2 e − 1 2 ( q k +1 − q k ) , k = 1 , ..., n − 1 b k = − 1 2 p k , k = 1 , ..., n . In these v ariables, the T o da system (2.3) b ecomes (2.5) da k dt = a k ( b k +1 − b k ) , k = 1 , ..., n − 1 db k dt = 2( a 2 k − a 2 k − 1 ) , k = 1 , ..., n with b oundary conditions a 0 = 0 , a n = 0 . This can b e written in Lax form as (2.6) d dt L ( t ) = [Sk ew( L ( t )) , L ( t )] with (2.7) L =       b 1 a 1 a 1 . . . . . . . . . . . . a n − 1 a n − 1 b n       , (2.8) Sk ew( L ) :=       0 a 1 − a 1 . . . . . . . . . . . . a n − 1 − a n − 1 0       . THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 5 An y equation in the Lax form d dt L = [ B , L ] for matrices L and B has the immediate consequence that the ﬂo w preserv es the sp ectrum of L . T o chec k this, it suﬃces to show that the functions tr( L k ) are constan t for all k . One shows ﬁrst by induction that d dt L k = [ B , L k ] and then observ es that d dt [tr( L k )] = tr[ d dt ( L k )] = tr[ B , L k ] = 0. W e now ha ve n − 1 indep endent in v ariant functions H k ( L ) = 1 k + 1 tr L k +1 . The Hamiltonian (2.1) is related to H 1 ( L ) b y H 1 ( L ) = 1 4 H ( p, q ) with the c hange of v ariables (2.4). If w e no w ﬁx the v alue of H 0 ( L ) = tr L (thus ﬁxing the momentum of the system), the resulting phase space has dimension 2( n − 1). With total momentum zero, this is (2.9) S =                  b 1 a 1 a 1 . . . . . . . . . . . . a n − 1 a n − 1 b n       : a i > 0 , b i ∈ R , n P i =1 b i = 0            . A property of real tridiagonal symmetric matrices (2.7) with a k 6 = 0 for all k is that the eigen v alues λ k are real and distinct. Let Λ be a set of n real distinct eigen v alues, and let M = { L ∈ S : sp ec( L ) = Λ } . Then S = ∪ Λ M Λ . S is in fact a symplectic manifold. Each in v ariant function H k ( L ) generates a Hamiltonian ﬂow via the symplectic structure, and the ﬂo ws are in volutiv e with resp ect to the symplectic structure (see [4] for the general framew ork and [29] for the T o da lattice sp eciﬁcally). W e will describ e the Lie-Poisson structure for the T o da lattice equations (2.6) in Section 2.5; how ev er, w e do not need the symplectic structure explicitly here. Moser [58] analyzes the dynamics of the T o da particles, showing that for an y initial conﬁguration, q k +1 − q k tends to ∞ as t → ±∞ . Thus, the oﬀ-diagonal en tries of L tend to zero as t → ±∞ so that L tends to a diagonal matrix whose diagonal en tries are the eigen v alues. W e will order them as λ 1 < λ 2 < · · · < λ n . The analysis in [58] shows that L ( ∞ ) = diag( λ n , λ n − 1 , · · · , λ 1 ) and L ( −∞ ) = diag( λ 1 , λ 2 , · · · , λ n ). The ph ysical in terpretation of this is that as t → −∞ , the particles q k approac h the velocities p k ( −∞ ) = − 2 λ k , and as t → ∞ , the velocities are interc hanged so that p k ( ∞ ) = − 2 λ n − k . Asymptotically , the tra jectories b ehav e as q k ( t ) ≈ λ ± k t + c ± k p k ( t ) ≈ λ ± k , where λ + k = λ k and λ − k = λ n − k . Symes solv es the T o da lattice using matrix factorization, the QR-factorization; his solution, whic h he veriﬁes in [77] and prov es in a more general context in [76] is equiv alent to the following. T o solve (2.5) with initial matrix L (0), tak e the exponential e tL (0) and use Gram-Sc hmidt orthonormalization to factor it as (2.10) e tL (0) = k ( t ) r ( t ) , where k ( t ) ∈ S O ( n ) and r ( t ) is upp er-triangular. Then the solution of (2.5) is (2.11) L ( t ) = k − 1 ( t ) L (0) k ( t ) = r ( t ) L (0) r − 1 ( t ) . Since the Gram-Schmidt orthonormalization of e tL (0) can b e done for all t , this sho ws that the solution of the T o da lattice equations (2.5) on the set (2.9) is deﬁned for all t . 6 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 W e also mention the notion of the τ -functions whic h pla y a k ey role of the theory of in tegrable systems (see for example [39]). Let us ﬁrst introduce the following symmetric matrix called the momen t matrix, (2.12) M ( t ) := e tL (0) T · e tL (0) = e 2 tL (0) = r T ( t ) k T ( t ) k ( t ) r ( t ) = r T ( t ) r ( t ) , where r T denotes the transpose of r , and note k T = k − 1 . The decomposition of a symmetric matrix to an upp er-triangular matrix times its transp ose on the left is called the Cholesky factorization. This factorization is used to ﬁnd the matrix r , and then the matrix k can b e found by k = e tL (0) r − 1 . The τ -functions, τ j for j = 1 , . . . , n − 1, are deﬁned by (2.13) τ j ( t ) := det ( M j ( t )) = j Y i =1 r i ( t ) 2 , where M j is the j × j upp er-left submatrix of M , and we denote diag ( r ) = diag ( r 1 . . . , r n ). Also note from (2.11), i.e. L ( t ) r ( t ) = r ( t ) L (0), that we ha ve a j ( t ) = a j (0) r k +1 ( t ) r k ( t ) . With (2.13), w e obtain (2.14) a j ( t ) = a j (0) p τ j +1 ( t ) τ j − 1 ( t ) τ j ( t ) , and from this w e can also ﬁnd the formulae for b j ( t ) of L ( t ) as b j ( t ) = 1 2 d dt ln  τ j ( t ) τ j − 1 ( t )  . One should note that the τ -functions are just deﬁned from the moment matrix M = e 2 tL (0) , and the solutions ( a j ( t ) , b j ( t )) are explicitly giv en by those τ -functions without the factorization. 2.2. Hessen b erg form. The symmetric matrix L in (2.7), when conjugated b y the diagonal matrix D = diag(1 , a 1 , . . . , a n − 1 ), yields a matrix Y = D LD − 1 in Hessen b erg form: (2.15) Y =       b 1 1 a 2 1 . . . . . . . . . . . . 1 a 2 n − 1 b n       , The T o da equations now tak e the Lax form (2.16) d dt Y = 2[ Y , F ] with F =       0 0 a 2 1 . . . . . . . . . . . . 0 a 2 n − 1 0       . Notice that Y ( t ) satisﬁes (2.16) if and only if X ( t ) = 2 Y ( t ) satisﬁes (2.17) d dt X = [ X , Π N − X ] , THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 7 where Π N − A is the strictly lo w er-triangular part of A obtained by setting all entries on and abov e the diagonal equal to zero. Equation (2.17) with (2.18) X =       f 1 1 g 1 . . . . . . . . . . . . 1 g n − 1 f n       is called the asymmetric, or Hessen b erg, form of the non-perio dic T o da lattice. Again, since the equations are in Lax form, the functions H k ( X ) = 1 k +1 tr X k +1 . are constan t in t . Notice that the Hessenberg and symmetric Lax form ulations of (2.3) are simply diﬀerent wa ys of expressing the same system. The solutions exist for all time and exibit the same behavior as t → ±∞ in b oth cases. How ever, when we generalize the T o da lattice to allo w the sub diagonal en tries to take on an y real v alue, the symmetric and Hessen b erg forms diﬀer in their geometry and top ology and in the character of their solutions. 2.3. Extended real tridiagonal symmetric form. Consider again the Lax equation (2.19) d dt L = [Skew( L ) , L ] , where w e now extend the set of matrices L of the form (2.7) b y allo wing a k to tak e on any real v alue. (Recall that in our original deﬁnition of the matrix L , each a k w as an exp onen tial and w as therefore strictly p ositive.) As b efore, b k ma y b e any real v alue, and Skew L is deﬁned as in (2.8). Giv en an y initial matrix in this extended form, the factorization metho d of Symes, describ ed in Section 2.1, still w orks. Indeed, for any suc h initial matrix L (0), e tL (0) can b e factored into (orthogonal) × (upp er-triangular) via the Gram-Schmidt procedure, i.e. the QR-factorization, and one can v erify as b efore that L ( t ) = k − 1 ( t ) L (0) k ( t ), where k ( t ) is the orthogonal factor. Th us, in the extended symmetric form, solutions are still deﬁned for all t . Given the initial eigenmatrix of L , an explicit solution of (2.19) can b e found in terms of the eigenmatrix of L (0) using the metho d of in verse scattering. F or a general Lax equation d dt Y = [ B , Y ], if Y (0) has distinct eigenv alues λ 1 , ..., λ n , then the in verse scattering scheme works as follows. Let Λ b e the diagonal matrix of eigenv alues, Λ = diag( λ 1 , ..., λ n ), and let Φ( t ) be a matrix of normalized eigenv ectors, v arying smo othly in t , where the k th column is a normalized eigenv ector of Y ( t ) with eigenv alue λ k . Then the Lax equation d dt Y = [ B , Y ] is the compatibilit y condition for the equations Y ( t )Φ( t ) = Φ( t )Λ (2.20) d dt Φ( t ) = B ( t )Φ( t ) . (2.21) W e see this as follows: Denoting ( · ) 0 = d dt ( · ), ( Y Φ) 0 = (ΦΛ) 0 ⇒ ( Y ) 0 Φ + Y (Φ) 0 = (Φ) 0 Λ ⇒ ( Y ) 0 Φ + Y B φ = B ΦΛ b y (2 . 21) ⇒ ( Y ) 0 Φ = − Y B Φ + B Y Φ by (2 . 20) ⇒ ( Y ) 0 Φ = [ B , Y ]Φ ⇒ ( Y ) 0 = [ B , Y ] . The in v erse scattering method solves the system (2.20) and (2.21) for φ ( t ) and then reco v ers Y ( t ) from (2.21). Since B ( t ) is deﬁned as a pro jection of Y ( t ), whic h can b e written in terms of Φ( t ) and Λ, we obtain a diﬀeren tial equation for Φ( t ) by replacing B ( t ) in (2.21) b y its expression in terms 8 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 + + +− − + −− (2,1,3) ( + + ) ( − −) ( − + ) ( + −) M M M M (1,2,3) (1,3,2) (3,1,2) (2,3,1) (3,2,1) Figure 2.1. The T omei manifold M Λ for the sl (3 , R ) T o da lattice. The 3-tuples ( i, j , k ) on the vertices indicate the diagonal matrices L = diag( λ i , λ j , λ k ). Eac h hexagon M  1 , 2 corresp onds to the moment polytop e (see Section 3.1) for the T o da lattice with the signs (  1 ,  2 ) = (sgn( a 1 ) , sgn( a 2 )). The b oundaries correspond to the sl (2 , R ) T o da lattices asso ciated with either a 1 = 0 or a 2 = 0. The T omei manifold M Λ is given by gluing edges of the hexagons. F or example, the edge b et ween (1 , 2 , 3) and (1 , 3 , 2) in M ++ is glued with the same edge in M − + , since this edge indicates a 1 = 0. The other gluing shown in the ﬁgure is for the edges corresp onding to a 2 = 0. of Φ( t ) and Λ. Given Y (0), we then obtain Φ(0) from (2.20) with t = 0. This solution is in fact equiv alent to the QR factorization giv en ab ov e (see Section 2.1). F or real matrices of the form (2.7), the in verse scattering metho d can b e used on the op en dense subset where all a k are nonzero. This is because a real matrix L of the form (2.7) has distinct real eigen v alues if a k 6 = 0 for all k . The eigenv alues are real b ecause L is a real symmetric matrix; the fact that they are distinct follows from the tridiagonal form with nonzero a k , which forces there to b e one eigenv ector (up to a scalar) for each eigen v alue. Let M Λ denote the set of n × n matrices of the form (2.7) with ﬁxed eigenv alues λ 1 < λ 2 < · · · < λ n . M Λ con tains 2 n − 1 comp onen ts of dimension n − 1, where each component consists of all matrices in M Λ with a ﬁxed choice of sign for eac h a k . The solution of (2.19) with initial condition in a given comp onen t remains in that component for all t ; that is, the solutions preserve the sign of eac h a k . Each lo wer-dimensional component, where one or more a k is zero and the signs of the other a k are ﬁxed, is also preserv ed b y the T o da ﬂow. Adding those low er dimensional comp onents giv es a compactiﬁcation of each comp onent of M Λ with ﬁxed signs in a k ’s. T omei [80] shows that M Λ is a compact smo oth manifold of dimension n − 1. In the pro of of this, he uses the T o da ﬂo w to construct co ordinate charts around the ﬁxed p oints. T omei sho ws that M Λ is orien table with univ ersal cov ering R n − 1 and calculates its Euler c haracteristic. In his analysis, T omei sho ws that M Λ con tains 2 n − 1 op en components diﬀeomorphic to R n − 1 . On eac h of these components, a k 6 = 0 for all k , and the sign of eac h a k is ﬁxed. They are glued together along the low er-dimensional sets where one or more a k is zero. F or example, in the case n = 3, there are four 2-dimensional components, denoted as M ++ , M + − , M − + , and M −− , according to the signs of a 1 and a 2 . The closure of eac h comp onen t is obtained by adding six diagonal matrices where all the a k v anish (these are the ﬁxed points of the T o da ﬂow) and six 1-dimensional sets where exactly one a k is zero. Denote the closure of M ++ b y M ++ , and so on. The boundary of M ++ , for example, con tains three 1-dimensional sets with a 1 = 0 and a 2 > 0. Each is c haracterized by having a ﬁxed eigen v alue as its ﬁrst diagonal entry . The other three 1-dimensional components in the boundary of THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 9 M ++ , hav e a 1 > 0 and a 2 = 0, with a ﬁxed eigen v alue in the third diagonal entry . Adding those b oundaries with the 6 vertices corresp onding to the diagonal matrices L = diag ( λ i , λ j , λ k ) giv es the compactiﬁed set M ++ . Notice that the three 1-dimensional comp onents with a 1 = 0 , a 2 > 0, and a ﬁxed λ k in the ﬁrst diagonal entry also lie along the b oundary of M − + ; the other three com p onen ts, with a 1 > 0 and a 2 = 0 are shared b y the b oundary of M ++ . In this manner, the four principal comp onen ts are glued together along the subset of M Λ where one or more a k v anish. In Figure 2.1, w e illustrate the compactiﬁcation of the T omei manifold M Λ for the sl (3 , R ) symmetric T o da lattice, M Λ = M ++ ∪ M + − ∪ M − + ∪ M −− , where the cups include the sp eciﬁc gluing according to the signs of the a k as explained ab ov e. The resulting compactiﬁed manifold M Λ is a connected sum of t w o tori, the compact Riemann surface of genus tw o. This can b e easily seen from Figure 2.1 as follo ws: Gluing those four hexagons, M Λ consists of 6 vertices, 12 edges and 4 faces. Hence the Euler characteristic is given by χ ( M Λ ) = 6 − 12 + 4 = − 2, whic h implies that the manifold has gen us g = 2 (recall χ = 2 − 2 g ). It is also easy to see that M Λ is orientable (this can b e sho wn b y giving an orientation for each hexagon so that the directions of tw o edges in the gluing cancel eac h other). Since the compact tw o dimensional surfaces are completely characterized by their orien tability and the Euler c haracters, w e conclude that the manifold M Λ is a connected sum of t wo tori, i.e. g = 2. The Euler characteristic of M Λ (for general n ) is determined in [80] as follows. Let L = diag( λ σ (1) , ..., λ σ ( n ) ) be a diagonal matrix in M Λ , where σ is a permutation of the n umbers { 1 , ..., n } , and let r ( L ) b e the n umber of times that σ ( k ) is less than σ ( k + 1). Denote by E ( n, k ) the n um b er of diagonal matrices in M Λ with r ( L ) = k . Then the Euler c haracteristic of M Λ is the alternating sum of the E ( n, k ): χ ( M Λ ) = n P k =0 ( − 1) k E ( n, k ) . [In M. Da vis et al extends T omei’s result....] If the eigenv alues of the tridiagonal real matrix L are not distinct, then one or more a k m ust b e zero. The set of suc h matrices with ﬁxed sp ectrum where the eigenv alues are not distinct is not a man- ifold. F or example, when n = 3 and the sp ectrum is (1 , 1 , 3), the isosp ectral set is one-dimensional since one a i is zero. It contains three diagonal matrices D 1 = diag(3 , 1 , 1), D 2 = diag(1 , 3 , 1), and D 3 = diag(1 , 1 , 3), and four 1-dimensional comp onents. Eac h 1-dimensional comp onent has a 1 in either the ﬁrst or last diagonal en try and a 2 × 2 blo ck on the diagonal with eigenv alues 1 and 3, where the oﬀ-diagonal en try is either p ositiv e or negativ e. The tw o comp onents with the 1 × 1 block in the last diagonal entry connect D 1 and D 2 , and the t wo components with the 1 × 1 blo ck in the ﬁrst diagonal entry connect D 2 and D 3 . In Figure 2.2, w e illustrate the isosp ectral set of those matrices whic h is singular with a shap e of ﬁgure eight. 2.4. Extended real tridiagonal Hessenberg form. W e now return to the Hessen berg form of the T o da equations with (2.22) X =       f 1 1 g 1 . . . . . . . . . . . . 1 g n − 1 f n       as in (2.18), and allow the g k to tak e on arbitrary real v alues. On the set of tridiagonal Hessen b erg matrices X with f k and g k real, the T o da ﬂow is deﬁned b y (2.23) d dt X ( t ) = [ X ( t ) , Π N − X ( t )] , 10 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 + + +− − + − − (1,1,3) ( 0 + ) M M M M (3,1,1) (1,3,1) ( + 0 ) (1,1,3) ( 0 − ) (3,1,1) (1,3,1) ( + 0 ) (1,1,3) ( 0 + ) (3,1,1) (1,3,1) ( − 0 ) (1,1,3) ( 0 − ) (3,1,1) (1,3,1) ( − 0 ) Figure 2.2. The isosp ectral set of the matrices having the eigenv alues { 1 , 1 , 3 } . Eac h polytop e M  1 , 2 con tains the set of matrices with either a 1 = 0 or a 2 = 0 and the signs  i = sgn( a i ). These p olytop es are obtained b y squeezing the p olytop es for the semisimple sl (3) T oda lattice in Figure 2.1 according to the degeneration of the eigen v alues 2 → 1. The gluing pattern is the same as in Figure 2.1, that is, identify , for example, the edge (0 +) in M ++ with the same one in M − + . The resulting v ariety is singular and has a shape of ﬁgure eight. as in (2.17). Recall that in the formulation of the original T o da equations, all the g k w ere p ositiv e, so that the eigen v alues w ere real and distinct. When g k 6 = 0 for some k , the eigenv alues may now b e complex or may coincide. W e will see that this causes blow-ups in the ﬂows so that the top ology of the isosp ectral manifolds are v ery diﬀerent from the top ology of the T omei manifolds described in the previous section. The matrices of the form (2.22) with g k 6 = 0 for all k are partitioned into 2 n − 1 diﬀeren t Hamil- tonian systems, eac h determined b y a c hoice of signs of the g k . Letting s k = ± 1 for k = 1 , ..., n and taking the sign of g k to be s k s k +1 , Ko dama and Y e [51] giv e the Hamiltonian for the system with this c hoice of signs as (2.24) H = 1 2 n P k =1 y 2 k + n − 1 P k =1 s k s k +1 e − ( x k +1 − x k ) , where f k = − 1 2 y k , k = 1 , ..., n g k = 1 4 s k s k +1 e − ( x k +1 − x k ) , k = 1 , ..., n − 1 . The system (2.23) is then called the indeﬁnite T o da lattice. The negative signs in (2.24) correspond to attractive forces b etw een adjacent particles, which causes the system to b ecome undeﬁned at ﬁnite v alues of t , as is seen in the solutions obtained by Kodama and Y e in [50] and [51] by in v erse scattering. The blow-ups in the solutions are also apparent in the factorization solution of the Hessen b erg form. T o solve (2.23) with initial condition X (0), factor the exp onential e tX (0) as (2.25) e tX (0) = n ( t ) b ( t ) , where n ( t ) is lo wer unipotent and b ( t ) is upper-triangular. Then, as shown b y [62] and [61], (2.26) X ( t ) = n − 1 ( t ) X (0) n ( t ) = b ( t ) X (0) b − 1 ( t ) THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 11 solv es (2.23). Notice that the factorization (2.25) is obtained b y Gaussian elimination, whic h multi- plies e tX (0) on the left b y elemen tary row operations to put it in upp er-triangular form. This pro cess w orks only when all principal determinants (the determinan ts of upp er left k × k blo c ks, which are the τ -functions deﬁned b elo w) are nonzero. A t particular v alues of t ∈ R , this factorization can fail, and the solution (2.26) b ecomes undeﬁned. The solutions ( f k , g k ) can b e expressed in terms of the τ -functions which are deﬁned b y (2.27) τ k ( t ) := det h ( e tX (0) ) k i = k Y j =1 d j ( t ) , where ( e tX (0) ) k is the k × k upp er-left submatrix of e tL (0) , and diag( b ) = diag( d 1 , . . . , d n ). With (2.26), w e hav e (2.28) g k ( t ) = g k (0) d k +1 ( t ) d k ( t ) = g k (0) τ k +1 ( t ) τ k − 1 ( t ) τ k ( t ) 2 . The function f k ( t ) are giv en by (2.29) f k ( t ) = d dt ln  τ k ( t ) τ k − 1 ( t )  . No w it it clear that the factorization (2.25) fails if and only if τ k ( t ) = 0 for some k . Then a blow-up (singularit y) of the system (2.23) can b e characterized b y the zero sets of the τ -functions. Example 2.1. T o see how blo w-ups o ccur in the factorization solution, consider the initial matrix X 0 =  1 1 − 1 − 1  . When t 6 = − 1, e tX 0 =  1 + t t − t 1 − t  =  1 0 − t 1+ t 1   1 + t t 0 1 1+ t  , and the solution evolv es as in (2.26). The τ -function is giv en b y τ 1 ( t ) = 1 + t , and when t = − 1, this factorization does not work. Ho w ever, w e can m ultiply e − 1 X 0 on the left by a low er unipotent matrix n − 1 (in this case the iden tity) to put it in the form wb , where w is a permutation matrix: e − 1 X 0 =  0 − 1 1 2  =  1 0 0 1   0 − 1 1 0   1 2 0 1  . This example will be tak en up again in Section 3.2, where it is sho wn how the factorization using a p erm utation matrix leads to a compactiﬁcation of the ﬂows. In general, when the factorization (2.25) is not p ossible at time t = ¯ t , e ¯ tX (0) can b e factored as e ¯ tX (0) = n ( ¯ t ) w b ( ¯ t ), where w is a p erm utation matrix. Ercolani, Flasc hk a, and Haine [25] use this factorization to complete the ﬂows (2.26) through the blow-up times by embedding them in to a ﬂag manifold. The details will be discussed in the next section, where w e consider the complex tridiagonal Hessen b erg form of the T o da lattice. Ko dama and Y e ﬁnd explicit solutions of the indeﬁnite T oda lattices b y in v erse scattering. Their metho d is used to solve a generalization of the full symmetric T o da lattice in [50] and is sp ecialized to the indeﬁnite tridiagonal Hessen b erg T o da lattice in [51]. F or the Hamiltonian (2.24), Ko dama and Y e make the c hange of v ariables (2.30) a k = 1 2 e − ( x k +1 − x k ) / 2 , k = 1 , ..., n − 1 , s k b k = − 1 2 y k , k = 1 , ..., n , 12 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 together with t → 1 2 t so that Hamilton’s equations tak e the form (2.31) da k dt = 1 2 a k ( s k +1 b k +1 − s k b k ) , db k dt = s k +1 a 2 k − s k − 1 a 2 k − 1 , with a 0 = a n = 0. Here w e switc hed the notation a k and b k from the original one in [50, 51]. This system is equiv alent to (2.23) with f k = s k b k and g k = s k s k +1 a 2 k . The system (2.31) can then be written in Lax form as (2.32) d dt ˜ L = [ ˜ B , ˜ L ] , where ˜ L is the real tridiagonal matrix (2.33) ˜ L =         s 1 b 1 s 2 a 1 0 · · · 0 s 1 a 1 s 2 b 2 s 3 a 2 · · · 0 . . . . . . . . . . . . . . . 0 · · · . . . s n − 1 b n − 1 s n a n − 1 0 · · · · · · s n − 1 a n − 1 s n b n         and ˜ B is the pro jection (2.34) ˜ B = 1 2 [( ˜ L ) > 0 − ( ˜ L ) < 0 ] . The in verse scattering sc heme for (2.34) is (2.35) ˜ L Φ = ΦΛ , d dt Φ = ˜ B Φ where Λ = diag( λ 1 , ..., λ n ) and Φ is the eigenmatrix of ˜ L , normalized so that (2.36) Φ S − 1 Φ T = S − 1 , Φ T S Φ = S with S = diag( s 1 , ..., s n ). Note that the matrix ˜ L of (2.33) is expressed as ˜ L = LS with the symmetric matrix L given b y (2.7) for the original T o da lattice. When S is the iden tity , (2.36) implies that Φ is orthogonal, and for S = diag(1 , ..., 1 , − 1 , ..., − 1), Φ is a pseudo-orthogonal matrix in O ( p, q ) with p + q = n . This then deﬁnes an inner pro duct for functions f and g on a set C , h f , g i := R C f ( λ ) g ( λ ) dµ ( λ ) = n P k =1 f ( λ k ) g ( λ k ) s − 1 k , with the indeﬁnite metric dµ := n P k =1 s − 1 k δ ( λ − λ k ) dλ and the eigenv alues λ k of ˜ L . Then the en tries of ˜ L can b e expressed in terms of the eigenv ector φ ( λ ) = ( φ 1 ( λ ) , . . . , φ n ( λ )) T , i.e. ˜ Lφ ( λ ) = λφ ( λ ), (2.37) ( ˜ L ) ij = s j h λφ i , φ j i = s j n P k =1 λ k φ i ( λ k ) φ j ( λ k ) s − 1 k . The explicit time evolution of Φ can be obtained using an orthonormalization procedure on functions of the eigenv ectors that generalizes the metho d used in [47] to solve the full symmetric T o da hierarch y . A brief summery of the pro cedure is as follo ws: First consider the factorization (called the HR- factorization), (2.38) e tX (0) = r ( t ) h ( t ) , THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 13 where r ( t ) is a low er triangular matrix and h ( t ) satisﬁes h T S h = S (if S = I , then h ∈ S O ( n ), i.e. the factorization is the QR-type). Then the eigenmatrix Φ( t ) = ( φ i ( λ j )) 1 ≤ i,j ≤ n is given by Φ( t ) = h ( t )Φ(0). No w one can write Φ( t ) as Φ( t ) = h ( t )Φ(0) = r − 1 ( t ) r ( t ) h ( t )Φ(0) = r − 1 ( t ) e tX (0) Φ(0) = r − 1 ( t )Φ(0) e t Λ . Since r ( t ) is low er triangular, this implies φ i ( λ, t ) = Span R  φ 0 1 ( λ ) e λt , . . . , φ 0 i ( λ ) e λt  , i = 1 , . . . , n . Then using the Gram-Sc hmidt orthogonalization method, the functions φ i ( λ, t ) can b e found as [51], (2.39) φ i ( λ, t ) = e λt p D i ( t ) D i − 1 ( t )          s 1 c 11 · · · s i − 1 c 1 ,i − 1 φ 0 1 ( λ ) s 1 c 21 · · · s i − 1 c 2 ,i − 1 φ 0 2 ( λ ) . . . . . . . . . . . . s 1 c i 1 · · · s i − 1 c i,i − 1 φ 0 i ( λ )          , where φ 0 i ( λ ) = φ 0 i ( λ, 0), c ij ( t ) = h φ 0 i , φ 0 j e λt i , and D k ( t ) = | ( s i c ij ( t )) 1 ≤ i,j ≤ k | . The solution of the in verse scattering problem (2.35) is then obtained from (2.39) using (2.37). The matrix ˜ M := ( c ij ) 1 ≤ i,j ≤ n is the moment matrix for the indeﬁnite T o da lattice which is deﬁned in the similar wa y as (2.12), i.e. ˜ M ( t ) := e t 2 ˜ L (0) S − 1 e t 2 ˜ L (0) T = Φ 0 e t Λ S − 1 Φ T 0 , where w e hav e used ˜ L (0)Φ 0 = Φ 0 Λ and Φ T 0 S Φ 0 = S . Then the τ -functions are deﬁned by (2.40) ˜ τ k ( t ) = det ( ˜ M k ( t )) = | ( c ij ( t )) 1 ≤ i,j ≤ k | = 1 s 1 · · · s k D k ( t ) . F rom (2.39) it follo ws that when ˜ τ k ( ¯ t ) = 0 for some k and time ¯ t , ˜ L ( t ) blows up to inﬁnity as t → ¯ t . In [51], Ko dama and Y e c haracterize the blo w-ups with the zeros of τ -functions and study the top ology of a generic isospectral set M Λ of the extended real tridiagonal T o da lattice in Hessenberg form. It is ﬁrst sho wn, using the T o da ﬂows, that b ecause of the blo w-ups in ˜ L , M Λ is a noncompact manifold of dimension n − 1. The manifold is then compactiﬁed b y completing the ﬂows through the blo w-up times. The 2 × 2 case is basic to the compactiﬁcation for general n . The set of 2 × 2 matrices with ﬁxed eigen v alues λ 1 and λ 2 , (2.41) M Λ = (  f 1 1 g 1 f 2  : λ 1 < λ 2 ) consists of t wo components, M + with g 1 > 0 and M − with g 1 < 0, together with tw o ﬁxed p oin ts, ˜ L 1 =  λ 1 1 0 λ 2  and ˜ L 2 =  λ 2 1 0 λ 1  . W riting f 2 = λ 1 + λ 2 − f 1 and substituting this into the equation for the determinant, f 1 f 2 − g 1 = λ 1 λ 2 , sho ws that M λ is the parab ola (2.42) g 1 = − ( f 1 − λ 1 )( f 1 − λ 2 ) . This parabola op ens down, crossing the axis g 1 = 0 at f 1 = λ 1 and f 1 = λ 2 , corresp onding to the ﬁxed points ˜ L 1 and ˜ L 2 . F or an initial condition with g 1 > 0, the solution is deﬁned for all t ; it ﬂo ws a wa y from p 2 to ward p 1 . This illustrates what is kno wn as the sorting prop ert y , which says that as t → ∞ , the ﬂow tends to w ard the ﬁxed point with the eigenv alues in decreasing order along the diagonal. The comp onent with g 1 < 0 is separated in to disjoin t parts, one with f 1 < λ 1 and the 14 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 + + +− − + −− ( + + ) ( − −) ( − + ) ( − −) ( + − ) ( + − ) ( − −) ( − −) ( − + ) ( + − ) ( − + ) 0 0 0 0 1 1 1 1 2 2 2 M M M M (1,2,3) (1,3,2) (2,1,3) (3,1,2) (3,2,1) (2,3,1) Figure 2.3. The compactiﬁcation of the isospectral manifold M Λ for the indeﬁnite sl (3 , R ) T o da lattice. As in Figure 2.1, eac h hexagon indicates the momen t polytop e asso ciated with the indeﬁnite T o da lattice. The signs (  1 ,  2 ) in M  1 , 2 are those of ( g 1 , g 2 ) for t → −∞ , and each sign in the hexagons indicates the signs of ( g 1 , g 2 ). The gluing rule according to the sign changes of g i is the same as that in the T omei manifold, but the pattern is no w diﬀeren t. F or example, the edge b et w een (2 , 3 , 1) and (3 , 2 , 1) in M −− is no w glued with that in M + − . The solid and dashed lines in the hexagons show the p oints where the solutions blow-up, i.e. τ k = 0 for k = 1 (solid) and for k = 2 (dashed). Also the n umbers in the hexagons indicate the n umber of blow-ups along the ﬂo w from t = −∞ to + ∞ (see Section 6.3). other with f 1 > λ 2 . The solution starting at an initial matrix with f 1 > λ 2 ﬂo ws to ward the ﬁxed p oin t ˜ L 2 as t → ∞ . F or an initial matrix with f 1 < λ 1 , the solution ﬂows aw ay from ˜ L 1 , blo wing up at a ﬁnite v alue of t . By adding a p oint at inﬁnity to connect these t wo branc hes of the parab ola, the ﬂo w is completed through the blow-up time and the resulting manifold is the circle, S 1 . F or general n , the manifold M Λ with sp ectrum Λ contains n ! ﬁxed p oints of the ﬂow, where the eigenv alues are arranged along the diagonal. These vertices are connected to each other by incoming and outgoing edges analogous to the ﬂows connecting the tw o v ertices in the case n = 2. On each edge there is one g k that is not zero. As in the case n = 2, edges in whic h the blow-ups o ccur are compactiﬁed by adding a p oint at inﬁnity . Ko dama and Y e then sho w ho w to glue on the higher-dimensional components where more than one g k is nonzero and compactify the ﬂo ws through the blo w-ups to produce a compact n -dimensional manifold. The result is nonorien table for n > 2. In the case n = 3, it is a connected sum of t wo Klein bottles. In Figure 2.3, we illustrate the compactiﬁcation of M Λ for the sl (3 , R ) indeﬁnite T o da lattice. With the gluing, the compactiﬁed manifold M Λ has the Euler c haracteristic χ ( M Λ ) = − 2 as in the case of the T omei manifold (see Figure 2.1). The non-orien tability can be sho wn by non-cancellation of the giv en orientations of the hexagons with this gluing. The compactiﬁcation was further studied by Casian and Ko dama [15] (also see [17]), where they show that the compactiﬁed isosp ectral manifold is iden tiﬁed as a connected completion of the disconnected Cartan subgroup of G = Ad ( S L ( n, R ) ± ). The manifold is diﬀeomorphic to a toric v ariety in the ﬂag manifold asso ciated with G . They also give a cellular decomp osition of the compactiﬁed manifold for computing the homology of the manifold. W e will show more details in Section 6.3, where the gluing rules are giv en b y the W eyl group action on the signs of the entries g j of the matrix X . THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 15 2.5. F ull Symmetric real T oda lattice. W e no w return to the symmetric T o da equation (2.43) d dt L = [ Skew( L ) , L ] as in (2.6), where L is now a full symmetric matrix with distinct eigenv alues. As in the tridiagonal case, Sk ew( L ) is the sk ew-symmetric summand in the decomp osition of L in to sk ew-symmetric plus lo wer-triangular. Deift, Li, Nanda, and T omei [22] sho w that (2.43) remains completely integrable ev en in this case. They present a suﬃcient n um b er of constan ts of motion in in v olution and construct the asso ciated angle v ariables. The phase space of the full symmetric real T o da lattice is the set of symmetric matrices, whic h w e denote Sym( n ) :=  L ∈ sl ( n, R ) : L T = L  . One can then deﬁne the Lie-Poisson structure (the Kostan t-Killilo v 2-form) on this phase space as follo ws. First w e deﬁne a nondegenerate inner product (the Killing form), h A, B i = tr( AB ), which iden tiﬁes the dual space sl ∗ ( n, R ) with sl ( n, R ). Then w e consider the Lie algebra splitting, sl ( n, R ) = B − ⊕ so ( n ) = B ∗ − ⊕ so ∗ ( n ) , where B − is the set of lo wer triangular matrices. With the inner pro duct, we iden tify B ∗ − ∼ = so ⊥ ( n ) = Sym( n ) , so ∗ ( n ) = B ⊥ − = N − , where N − is the set of strictly low er triangular matrices, and A ⊥ indicates the orthogonal comple- men t of A with resp ect to the inner pro duct. The Lie-P oisson structure is then deﬁned as follows: F or any functions f and h on B ∗ − = Sym( n ), deﬁne { f , h } ( L ) = h L, [Π B − ∇ f , Π B − ∇ h ] i , where h X, ∇ f i = lim  → 0 d d f ( L + X ), and Π B − X is the pro jection of X on to B − . The T o da lattice (2.43) can no w b e expressed in Hamiltonian form as d dt L = { H 1 , L } ( L ) with H 1 ( L ) = 1 2 tr( L 2 ) . Using the P oisson structure, w e can no w extend equation (2.43) to deﬁne the T o da lattice hierarc h y generated b y the Hamiltonians H k ( L ) = 1 k +1 tr( L k +1 ): (2.44) ∂ ∂ t k L = { H k , L } ( L ) = [Skew( L k ) , L ] with H k = 1 k + 1 tr( L k +1 ) , where Sk ew( L k ) = Π so ( n ) ∇ H k . The ﬂow sta ys on a co-adjoin t orbit in Sym( n ) ∼ = B ∗ − . The Lie- P oisson structure is nondegenerate when restricted to the co-adjoint orbit, and the lev el sets of the in tegrals found in [22] are the generic co-adjoint orbits. Deift and colleagues ﬁnd the constants of motion by taking the matrices ( L ) k obtained by removing the ﬁrst k rows and last k columns of L . A co-adjoint orbit is obtained by ﬁxing the trace of each ( L ) k . The remaining co eﬃcients of the characteristic p olynomials of ( L ) k for 0 ≤ k ≤ [ n/ 2] (that is, all coeﬃcients except for the traces) pro vide a family of [ n 2 / 4] constan ts of motion in inv olution on the orbit. Generically , ( L ) k has n − 2 k distinct eigen v alues λ 1 ,k , ..., λ n − 2 k,k . The constants of motion ma y be tak en equiv alently as the eigenv alues λ r,k for 0 ≤ k ≤ [ 1 2 ( n − 1)] and 1 ≤ r ≤ n − 2 k . In this case the associated angle v ariables are essen tially the last comp onen ts of the suitably normalized eigen vectors of the ( L ) k . In [47], Kodama and McLaughlin give the explicit solution of the T o da lattice hierarch y (2.44) on full symmetric matrices with distinct eigen v alues b y solving the inv erse scattering problem of the 16 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 system L Φ = Φ Λ , ∂ ∂ t k Φ = Sk ew( L k ) Φ . with Λ = diag ( λ 1 , . . . , λ n ). Since L is symmetric, the matrix Φ of eigenv ectors is tak en to b e orthogonal: L = Φ Λ Φ T with Φ = [ φ ( λ 1 ) , ..., φ ( λ n )], where the φ ( λ k ) is the normalized eigen vector of L with eigen v alue λ k . The indeﬁnite extension of the full symmetric T o da lattice (where ˜ L = LS as in (2.33) is studied in [47], where explicit solutions of φ ( λ k , t ) are obtained b y inv erse scattering. The authors also give an alternativ e deriv ation of the solution using the factorization metho d of Symes [77], where e tL (0) is factored into a product of a pseudo-orthogonal matrix times an upp er triangular matrix as in (2.38) (the HR-factorization). 3. Complex Toda la ttices Here we consider the iso-sp ectral v arieties of the complex T o da lattices. In order to describ e the geometry of the iso-sp ectral v ariet y , we ﬁrst give a summary of the moment map on the ﬂag manifold. The general description of the momen t map discussed here can b e found in [44, 34]. 3.1. The momen t map. Let G b e a complex semisimple Lie group, H a Cartan subgroup of G , and B a Borel subgroup con taining H . If P is a parabolic subgroup of G that contains B , then G/P can b e realized as the orbit of G through the pro jectivized highest weigh t vector in the pro jectivization, P ( V ), of an irreducible representation V of G . Let A b e the set of weigh ts of V , counted with m ultiplicity; the w eigh ts belong to H ∗ R , the real part of the dual of the Lie algebra H of H . Let { v α : α ∈ A} b e a basis of V consisting of w eight vectors. A point [X] in G/P , represented by X ∈ V , has homogeneous coordinates π α ( X ), where X = P α ∈A π α ( X ) v α . The momen t map as deﬁned in [44] sends G/P in to H ∗ R : (3.1) µ : G/P − → H ∗ R [ X ] 7− → P α ∈A | π α ( X ) | 2 α P α ∈A | π α ( X ) | 2 Its image is the w eight polytop e of V , also referred to as the moment polytop e of G/P . The ﬁxed p oints of H in G/P are the p oints in the orbit of the W eyl group W through the pro jectivized highest weigh t v ector of V ; they corresp ond to the vertices of the p olytop e under the momen t map. Let H · [ X ] b e the closure of the orbit of H through [ X ]. Its image under µ is the con vex hull of the v ertices corresp onding to the ﬁxed points con tained in H · [ X ]; these v ertices are the weigh ts { α ∈ W · α V : π α ( X ) 6 = 0 } , where α V is the highest weigh t of V [5]. In particular, the image of a generic orbit, where no π α v anishes, is the full p olytop e. The real dimension of the image is equal to the complex dimension of the orbit. In the case that P = B , V is the representation whose highest weigh t is the sum of the fundamental w eights of G , whic h we denote as δ . Let v δ b e a weigh t vector with w eight δ . Then the action of G through [ v δ ] in P ( V ) has stabilizer B so that the orbit G · [ v δ ] is iden tiﬁed with the ﬂag manifold G/B . The pro jectivized weigh t v ectors that b elong to G/B are those in the orbit of the W eyl group, W = N ( H ) /H , through [ v δ ], where N ( H ) is the normalizer of H in G . These are the ﬁxed points THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 17 2130 3120 3210 2310 2301 3021 2031 1032 1023 0123 0213 1203 1302 3012 3102 3201 2130 2103 Figure 3.1. The moment polytop e of G/B for G = S L (4 , C ). The num b er set i 1 i 2 i 3 i 4 indicates the weigh t L = i 1 L 1 + i 2 L 2 + i 3 L 3 + i 4 L 4 . The highest weigh t corresp onds to the set 3210. of H in G/B . The stabilizer in W of [ v δ ] is trivial so that in G/B , the ﬁxed points of H are in bijection with the elemen ts of the W eyl group. No w take G = S L ( n, C ), B the upp er triangular subgroup, and H the diagonal torus. The c hoice of B determines a splitting of the root system into positive and negative roots and a system ∆ of simple ro ots. The simple roots are L i − L i +1 , where i = 1 , ..., n − 1 and L i is the linear function on H that gives the i th diagonal entry; the W eyl group is the permutation group Σ n , whic h acts b y p erm uting the L i . H ∗ R is the quotien t of the real span of the L i b y the relation L 1 + · · · + L n = 0. H ∗ R ma y be viewed as the h yp erplane in R n where the sum of the co eﬃcients of the L i is equal to 1 + 2 + · · · + ( n − 1). Let { i 1 , ..., i n } = { 0 , ..., n } . The momen t polytop e is the con vex h ull of the w eights L = i 1 L 1 + i 2 L 2 + · · · + i n L n , where ( i 1 , i 2 , . . . , i n − 1 , i n ) = ( n − 1 , n − 2 , · · · , 1 , 0) corresp onds to the highest weigh t. In Figure 3.1, w e illustrate the moment p olytop e for the ﬂag manifold G/B of G = S L (4 , C ). Let S be the set of reﬂections of H ∗ R in the h yp erplanes perp endicular, with respect to the Killing form, to the simple ro ots (these are the simple reﬂections). The group of motions of H ∗ R generated b y S is isomorphic to W ; it is also denoted as W and referred to as the W eyl group of G . The v ertices of the moment p olytop e are the orbit of W through δ . F or w ∈ W , the momen t map µ sends w [ v δ ] to the vertex w ( δ ). An arbitrary w ∈ W can be written as a comp osition w = s r . . . s 1 of simple reﬂections s i . The length , l ( w ), of w with resp ect to the simple system ∆ is the smallest r for whic h such an expression exists. 3.2. Complex tridiagonal Hessen b erg form. Here w e consider the set M of complex tridiagonal Hessen b erg matrices (3.2) X =       f 1 1 g 1 . . . . . . . . . . . . 1 g n − 1 f n       , where the f k and g k are allow ed to b e arbitrary complex num b ers. As b efore, the T o da ﬂow is deﬁned by (2.23) and the eigen v alues (equiv alen tly , the traces of the pow ers of X ) are constan ts of motion. The Hamiltonian H k ( X ) = 1 k + 1 tr  X k +1  18 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 generates the ﬂo w (3.3) ∂ X ∂ t k = [ X ( t k ) , Π N − ( X k ( t k ))] , b y the Poisson structure on M that we will deﬁne in Section (3.3). This gives a hierarch y of comm uting independent ﬂows for k = 1 , ..., n − 1. F rom this w e can see that the trace of X is a Casimir with trivial ﬂow. The solution of (3.3) can b e found by factorization as in (2.25): F actor e t k X k (0) as (3.4) e t k X k (0) = n ( t k ) b ( t k ) , where n ( t k ) is lo wer unipotent and b ( t k ) is upp er-triangular. Then (3.5) X ( t k ) = n − 1 ( t k ) X (0) n ( t k ) solv es (3.3). 3.2.1. Char acterization of blow-ups via Bruhat de c omp osition of G/B . Fix the eigen v alues λ j , and consider the lev el set M Λ consisting of all matrices in M with sp ectrum ( λ 1 , ..., λ n ). In the case of distinct eigenv alues, Ercolani, Flasc hk a, and Haine [25] construct a minimal nonsingular compacti- ﬁcation of M Λ on whic h the ﬂo ws (3.3) extend to global holomorphic ﬂows. The compactiﬁcation is induced b y an em b edding of M Λ in to the ﬂag manifold G/B , where G = S L ( n, C ) and B is the upp er triangular subgroup of G . Prop osition 3.1. (Kostant, [54] ) Consider the matrix (3.6)  Λ =         λ 1 1 0 · · · 0 0 λ 2 1 · · · 0 . . . . . . . . . . . . . . . 0 · · · . . . λ n − 1 1 0 · · · · · · 0 λ n         in  Λ . Every X ∈ M Λ c an b e c onjugate d to  Λ by a unique lower-triangular unip otent matrix L : (3.7) X = L Λ L − 1 . This deﬁnes a map of M Λ in to G/B : (3.8) j Λ : M Λ → G/B X 7→ L − 1 mo d B . This mapping is an embedding [53], and the closure, j Λ ( M Λ ), of its image is a nonsingular and minimal compactiﬁcation of M Λ [25]. Let L 0 b e the unique low er unip otent matrix such that X (0) = L 0  Λ L − 1 0 . Then the solution (3.5) is conjugate to  Λ as X ( t k ) = n − 1 ( t k ) L 0  Λ L − 1 0 n ( t k ), where L − 1 0 n ( t k ) is lo wer unipotent. Thus, X ( t k ) is mapp ed into the ﬂag manifold as j Λ ( X ( t k )) = L − 1 0 n ( t k ) mo d B (3.9) = L − 1 0 e t k X k mo d B . (3.10) Notice that even at v alues of t k where (3.10) is not deﬁned b ecause the factorization (3.4) is not p ossible, the equiv alent expression (3.10) is deﬁned. In this wa y , the em b edding of X ( t k ) in to G/B completes the ﬂo ws through the blo w-up times. This giv es a compactiﬁcation of M Λ in G/B . [25] uses this em b edding to study the nature of the blow-ups of X ( t k ). THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 19 T o illustrate this in a simple case, consider Example 1.1 from the Section 2.4. The isosp ectral set of 2 × 2 Hessen b erg matrices with b oth eigenv alues zero is embedded into the ﬂag manifold S L (2 , C ) /B , which has the cell decomposition (3.11) S L (2 , C ) /B = N B /B ∪ N  0 − 1 1 0  B /B . Here N is the subgroup of lo w er unipotent matrices. The big cell, N B /B , contains the image of the ﬂow X ( t ) whenev er this ﬂow is deﬁned, that is, whenev er the factorization e tX 0 = n ( t ) b ( t ) is p ossible. At t = − 1, where X ( t ) is undeﬁned, the embedding j Λ completes the ﬂow through the singularit y . The image j Λ ( X ( t )) passes through the ﬂag L − 1 0 e − 1 X (0) at time t = − 1, which is the cell on the righ t in (3.11). The cell decomp osition (3.11) is a sp ecial case of the cell stratiﬁcation of the ﬂag manifold G/B kno wn as the Bruhat decomp osition. This decomp osition is deﬁned in terms of the W eyl group, W , as (3.12) G/B = [ w ∈ W N wB /B . In the presen t case of G = S L ( n, C ), W is essentially the group of p ermutation matrices. Th us, the Bruhat decomp osition partitions ﬂags according to whic h permutation matrix w is needed to p erform the factorization X = nwb for X ∈ G with n ∈ N and b ∈ B . A t all v alues of t k for which the ﬂo w X ( t k ) is deﬁned, j Λ sends X ( t k ) in to the big cell of the Bruhat decomp osition, since w is the iden tity . When the factorization (3.4) is not p ossible at time t k = ¯ t , e ¯ tX k (0) can b e factored as (3.13) e ¯ tX k (0) = n ( ¯ t ) w b ( ¯ t ) , for some permutation matrix w . In this case, the ﬂow (3.10) enters the Bruhat cell N w B /B at time t k = ¯ t . [25] c haracterizes the Laurent expansion of eac h p ole of X ( t 1 ) in terms of the Bruhat cell that the solution en ters at the blow-up time. It is also seen in [25] that for k = 1 , ..., n − 1, the ﬂo ws (3.10) generate a ( C ∗ ) n − 1 torus action on G/B that has trivial isotrop y group at ev ery j Λ ( X ) with g k ( X ) 6 = 0 for all k . The orbit through an y suc h point is op en and dense in j Λ ( M Λ ), and the closure of this orbit is the minimal compactiﬁcation of j Λ ( M Λ ) in G/B . 3.2.2. Comp actiﬁc ation of iso-level set with arbitr ary sp e ctrum. Here M and M Λ are again deﬁned as in Section 3.2. Shipman [72] uses a diﬀeren t embedding, referred to as the Jordan em bedding, of M Λ in to G/B to describ e the compactiﬁcation of an isosp ectral set M Λ with arbitrary sp ectrum. The adv antage of the Jordan embedding is that the maximal torus generated by the ﬂo ws is diagonal if the eigenv alues are distinct and a product of a diagonal torus and a unip otent group when eigenv alues coincide. The orbits of these groups, sp eciﬁcally the torus comp onent, are easily studied by taking their images under the moment map, as explained b elow. This leads to a simple description of the closure of M Λ in terms of faces of the momen t p olytop e. Recall that in the real tridiagonal Hessen b erg form of the T o da lattice studied by Kodama and Y e [51] (see Section 2.4), the ﬂows through an initial matrix X preserve the sign of eac h g k that is not zero and preserv e the v anishing of eac h g k that is zero. The op en subset of the isospectral set where no g k v anishes is partitioned in to 2 n − 1 comp onen ts, according to the signs of the g k . The compactiﬁcation of the isosp ectral set is obtained b y completing the ﬂows through the blo w-up times and pasting the 2 n − 1 comp onen ts together along the lo wer-dimensional pieces where one or more g k v anishes, pro ducing a compact manifold. In con trast to this, when X is complex, M Λ is no longer partitioned b y signs of the g k ; there is only one maximal comp onent where no g k v anishes. The n − 1 ﬂows through an y initial X with g k 6 = 0 for all k generates the whole component, as w as also observ ed in [25]. 20 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 T o deﬁne the Jor dan emb e dding , let C Λ b e the companion matrix of X , (3.14) C Λ =        0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . . . . 0 0 · · · 0 1 s n s n − 1 · · · s 2 0        . Here the s j ’s are the symmetric p olynomials of the eigev alues λ j , i.e. det( λI − X ) = λ n − n P j =2 s j λ n − j . Again, b y [54], there exists a unique lo wer unipotent matrix L such that X = LC Λ L − 1 . In particular, all elemen ts of M Λ are conjugate. Since the companion matrix has a single chain of generalized eigen vectors for each eigen v alue, any matrix in Jordan canonical form that is conjugate to it contains one blo c k for each eigen v alue. F ollowing [70], ﬁx an ordering of the eigen v alues, and let J b e the corresp onding Jordan matrix. Then C = W J W − 1 where W is a matrix whose columns are (generalized) eigen v ectors of C , where eac h eigenv ector has a 1 in the ﬁrst nonzero entry and the vectors are ordered according to the c hosen ordering of eigenv alues, with generalized eigen vectors ordered successively . Once W is ﬁxed, for X ∈ M Λ , w e can write X = LW J W − 1 L − 1 . The Jordan em b edding is the mapping (3.15) γ Λ : M Λ − → G/B X 7− → W − 1 L − 1 mo d B That this is an em b edding follows from the results in [53]. Under this em bedding, the ﬂows X ( t k ) = n − 1 ( t k ) X (0) n ( t k ) in (3.5) with e t k X k − 1 (0) = n ( t k ) b ( t k ) as in (3.4) and X (0) = LW J W − 1 L − 1 generate a group action as follo ws: γ Λ ( X ( t k )) = W − 1 L − 1 n k ( t k ) mo d B = W − 1 L − 1 exp[ t k X k (0) ] mo d B = W − 1 L − 1 exp[ t k ( LW J Λ W − 1 L − 1 ) k ] mo d B = W − 1 L − 1 L W exp[ t k J k Λ ] W − 1 L − 1 mo d B = exp[ t k J k Λ ] W − 1 L − 1 mo d B The ﬂows exp[ t k J k Λ ] for k = 1 , ..., n − 1 generate the centralizer of J in S L ( n, C ). W e denote this subgroup as A J . A J has r blocks along the diagonal, (3.16) A i =                  s x 1 · · · x d i − 1 . . . . . . . . . . . . x 1 s       : s ∈ C ∗ , x 1 , ..., x d i − 1 ∈ C            , i = 1 , . . . , r , where d i is the m ultiplicity of the eigenv alue in the i th block of J . All the blocks together con tain n − r independent en tries in C ab o ve the diagonal and r en tries in C ∗ on the diagonal, where the pro duct of the diagonal en tries is 1. A J is a semi-direct pro duct of the diagonal torus K J , obtained by setting all the en tries ab ov e the diagonal equal to zero, and the unip otent group U J , obtained b y setting all the diagonal entries equal to 1. The subgroup of A J that ﬁxes ev ery p oint in G/B is the (discrete) subgroup D of all constan t m ultiples of the iden tity . The quotient A J /D THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 21 has the manifold structure (but not the group structure) of ( C ∗ ) r − 1 × C n − r . When r = n (distinct eigen v alues), A J is the maximal diagonal torus. The compactiﬁcation, M Λ , of M Λ in G/B is the closure of one generic orbit of A J . Its boundary is a union of non-maximal orbits of A J . [72] uses the momen t map of the maximal torus action, whic h sends S L ( n, C ) /B to a p olytop e in R n − 1 , to iden tify eac h comp onent of the b oundary of M Λ with a speciﬁed face of the p olytop e, as describ ed ab o ve. First w e describe the b oundary of M Λ in M . Let α b e a subset of { g 1 , ..., g n − 1 } , and denote b y M α Λ the subset of M Λ on which exactly the g i in α are zero. These subsets form a partition of M Λ where the complex dimension of M α Λ is equal to the num b er of g i that do not v anish. There is one maximal comp onent, on whic h no g k v anish, and one comp onent consisting of the ﬁxed points, where all the g k v anish. Let X ∈ M α Λ . The blo cks on the diagonal of X where no g k v anish are full tridiagonal Hessen b erg matrices of a smaller dimension (all en tries on their ﬁrst subdiagonals are nonzero). The union of the eigen v alues of these blo c ks, coun ted with multiplicit y , is the sp ectrum Λ. Let P (Λ) be a partition of Λ into subsets Λ = Λ 1 ∪ ... ∪ Λ q , where Λ k is the spectrum of the k th block along the diagonal of X , and denote b y M α P (Λ) the component of M α Λ where Λ is partitioned among the blo cks according to P (Λ). The T o da ﬂo ws (3.5) through X preserve the sp ectrum of eac h block and therefore respect the partition M Λ = ∪M α P (Λ) . The momen t map, described abov e, gives a one-to-one correspondence b et ween the components M α P (Λ) and particular faces of a certain p olytop e [72]. T o see this, let K J b e the torus that lies along the diagonal of A J . K J is a subtorus of the maximal diagonal torus H . Its Lie algebra, K J , is the kernel of a subset ∆ J of the simple ro ots; this determines the subset S J ⊂ S of reﬂections in the hyperplanes p erp endicular to the roots in ∆ J . S J generates a subgroup W J of W . The elements in W J = { w ∈ W : l ( sw ) > l ( w ) ∀ s ∈ S J } are the coset represen tativ es of minim um length in the quotient W J \ W . The following result is prov ed in [72]: Prop osition 3.2. [72] The c omp osition µ ◦ γ Λ : M Λ → H ∗ R gives a one-to-one c orr esp ondenc e b etwe en the c omp onents M α P (Λ) that p artition M Λ and the fac es of the moment p olytop e with at le ast one vertex in W J . The c omplex dimension of the c omp onent is e qual to the r e al dimension of the fac e. In p articular, the maximal orbit in M Λ c orr esp onds to the ful l p olytop e, and the ﬁxe d p oints of A J in M Λ c orr esp ond to the vertic es in W J . 3.3. The full Kostant-T oda lattice. Here we consider full complex Hessen b erg matrices (3.17) X =        ∗ 1 0 · · · 0 ∗ ∗ 1 · · · 0 . . . . . . . . . . . . . . . ∗ ∗ ∗ · · · 1 ∗ ∗ ∗ · · · ∗        . The set of all such X is denoted  + B − , where  is the matrix with 1’s on the superdiagonal and zeros elsewhere and B − is the set of lo wer triangular complex matrices. With respect to the symplectic structure on  + B − deﬁned below, the T o da hierarc hy (3.3) with X as in (3.17) turns out to be completely in tegrable on the generic leav es. The complete integrabilit y is observed in [26] b y extending the results of [22] to  + B − . F or n > 3, the eigenv alues of the initial matrix do not constitute enough integrals for complete integrabilit y; a generic level set of the constants of motion is a subset of an isosp ectral set that is cut out by additional integrals and Casimirs, whic h can b e computed by a c hopping construction given in Proposition (3.3). 22 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 Ik eda [40] studies the lev el sets in  + B − cut out b y ﬁxing only the eigen v alues. He ﬁnds a compactiﬁcation of an isospectral set with distinct eigen v alues, sho wing that its cohomology ring is the same as that of the ﬂag manifold S L ( n, C ) /B . This w ork diﬀers from the previous works [80], [72], and [51] that compactify tridiagonal versions of the T o da lattice in that it do es not use the T o da ﬂows directly in producing the compactiﬁcation. T o describ e the symplectic structure on  + B − , write (3.18) sl ( n, C ) = N − ⊕ B + , where N − and B + are the strictly low er triangular and the upp er triangular subalgebras, resp ec- tiv ely . With a non-degenerate inner product h A, B i = tr( AB ) on sl ( n, C ), we ha v e an isomorphism sl ( n, C ) ∼ = sl ∗ ( n, C ) and sl ∗ ( n, C ) = N ∗ − ⊕ B ∗ + = B ⊥ + ⊕ N ⊥ − . With the isomorphisms B ∗ + ∼ = N ⊥ − = B − , N ∗ − ∼ = B ⊥ + = N + w e identify  + B − ∼ = B ∗ + , whic h deﬁnes the phase space of the full Kostant-T oda lattice. On the space B ∗ + , we deﬁne the Lie-P oisson structure (Kostant-Kirillo v form); that is, for any functions f , h on B ∗ + , { f , h } ( X ) = h L, [Π B + ∇ f , Π B + ∇ h ] i , for X ∈ B ∗ + , where h Y , ∇ f i = lim  → 0 d d f ( X + Y ). This Lie-P oisson structure gives a stratiﬁcation of B ∗ + ∼ =  + B − . The stratiﬁcation of the P oisson manifold B ∗ + with this Lie-P oisson structure is complicated, ha ving lea ves of diﬀeren t types and diﬀerent dimensions. Denote by B + the upp er-triangular subgroup of S L ( n, C ) and by g · Y the adjoint action of S L ( n, C ) on sl ( n, C ). Then, through the identiﬁcation of  + B − with B ∗ + , the abstract coadjoin t action of B + on B ∗ + b ecomes Ad ∗ b X =  + Π B − b − 1 · ( X −  ) . The symplectic lea ves in  + B − are generated b y the coadjoint orbits and additional Casimirs. In general, the dimension of a generic leaf is greater than 2( n − 1), and more in tegrals are needed for complete integrabilit y . The c hopping construction used in [22] to obtain a complete family of in tegrals for the full symmetric T o da lattice is adapted in [26] to ﬁnd a complete family of integrals for the full asymmetric T o da lattice. Prop osition 3.3. [26] Cho ose X ∈  + B − , and br e ak it into blo cks of the indic ate d sizes as X =   k n − 2 k k k X 1 X 2 X 3 n − 2 k X 4 X 5 X 6 k X 7 X 8 X 9   , wher e k is an inte ger such that 0 ≤ k ≤ [ ( n − 1) 2 ] . If det( X 7 ) 6 = 0 , deﬁne the matrix φ k ( X ) by φ k ( X ) = X 5 − X 4 X − 1 7 X 8 ∈ Gl ( n − 2 k , C ) , k 6 = 0 , φ 0 ( X ) = X. The c o eﬃcients of the p olynomial det( λ − φ k ( X )) = λ n − 2 k + I 1 k λ n − 2 k − 1 + · · · + I n − 2 k,k ar e c onstants of motion of the ful l Kostant-T o da lattic e. The functions I 1 k ar e Casimirs on  + B − , and the functions I rk for r > 1 c onstitute a c omplete involutive family of inte gr als for the generic symple ctic le aves of  + B − cut out by the Casimirs I 1 k . These inte gr als ar e known as the k -chop inte gr als. THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 23 The k -chop in tegrals I rk are equiv alen t to the traces of the p o wers of φ k ( X ). The Hamiltonian system generated b y an integral I ( X ) is (3.19) d dt X = [ X , Π N − ( ∇ I ( X ))] . When I ( X ) is one of the original T oda in v ariants H k ( X ) = 1 k +1 tr( X k + 1) (a 0-chop in tegral), the ﬂo w is (3.20) d dt X = [ X , Π N − ( X k )] . The solution ma y again b e found via factorization [26]. Let e t ∇ I ( X 0 ) = n ( t ) b ( t ) , with n ( t ) and b ( t ) lo wer unipotent and upper-triangular, resp ectively . Then X ( t ) = n − 1 ( t ) X 0 n ( t ) . Let X b elong to (  + B − ) Λ . Recall from Section 3.2 that there exists a unique lo wer unipotent matrix L suc h that X = LC Λ L − 1 , where C is the companion matrix (3.14). The mapping (3.21) c Λ : M Λ − → S L ( n, C ) /B X 7− → L − 1 mo d B is an em b edding [53], referred to as the c omp anion emb e dding . Its image is open and dense in the ﬂag manifold. Under this em b edding, the n − 1 ﬂo ws of the 0-chop integrals 1 k tr X k generate the action of the cen tralizer of C Λ in S L ( n, C ) (the group acts by m ultiplication on the left). When the λ i are distinct, C Λ = V Λ V − 1 , where V is a V andermonde matrix, and X = LV Λ V − 1 L − 1 . The em b edding (3.22) Ψ Λ : (  + B − ) Λ − → S L ( n, C ) /B X 7− → V − 1 L − 1 mo d B is a sp eciﬁc case of the Jordan em b edding (3.15) when the eigenv alues are distinct. In this case, the group A J (see (3.16)) generated b y Hamiltonian ﬂo ws of H k = 1 k +1 tr( X k ) for k = 1 , ..., n − 1 is the maximal diagonal torus. Ψ Λ is therefore referred to as the torus emb e dding . When the v alues of the integrals are suﬃciently generic (in particular, when the eigenv alues of eac h k -chop are distinct), Ercolani, Flaschk a, and Singer [26] sho w how the ﬂows of the k -chop in tegrals can b e organized in the ﬂag manifold by the torus embedding. (The companion em b edding giv es a similar structure, but the torus embedding is more conv enient since the group action is diagonal.) The guiding idea in [26] is that the k -chop in tegrals for S L ( n, C ) are equiv alent to the 1-c hop in tegrals for S L ( n − 2( k − 1) , C ). Let S L ( m, C ) /B denote the quotient of S L ( m, C ) by its upp er triangular subgroup, and let S L ( m, C ) /P denote the quotien t of S L ( m, C ) b y the parab olic subgroup P of S L ( m, C ) whose entries below the diagonal in the ﬁrst column and to the left of the diagonal in the last ro w are zero: P =        ∗ ∗ · · · ∗ ∗ 0 ∗ · · · ∗ ∗ . . . . . . . . . . . . . . . 0 ∗ · · · ∗ ∗ 0 0 · · · 0 ∗        . 24 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 The 1-c hop in tegrals I r 1 dep end only on the partial ﬂag manifold S L ( n, C ) /P . In this partial ﬂag manifold, a level set of the 1-c hop integrals is generated b y the ﬂo ws of the 0-chop torus. The 1-c hop ﬂo ws generate a torus action along the ﬁb er of the pro jection S L ( n − 2 , C ) /B → S L ( n, C ) /B ↓ S L ( n, C ) /P . In this ﬁber, the 2-chop integrals dep end only on the partial ﬂag manifold S L ( n − 2 , C ) /P , where a level set of the 2-chop in tegrals is generated b y the 1-c hop torus. This picture extends to all the k -chop ﬂo ws. [26] builds a tow er of ﬁbrations S L ( n − 2( k + 1) , C ) /B → S L ( n − 2 k, C ) /B ↓ S L ( n − 2 k , C ) /P where the k -chop ﬂo ws generate a lev el set of the ( k + 1)-chop in tegrals in the partial ﬂag manifold S L ( n − 2 k , C ) /P and the ( k + 1)-ﬂows act as a torus action along the ﬁb er, S L ( n − 2( k + 1) , C ) /B . In the end, the closure of a level set of all the k -chop in tegrals in S L ( n, C ) /B is realized as a pro duct of closures of generic torus orbits in the pro duct of partial ﬂag manifolds (3.23) S L ( n, C ) /P × S L ( n − 2 , C ) /P × · · · × S L ( n − 2 M , C ) /P where M is largest k for whic h there are k -chop in tegrals. In [33], Gekh tman and Shapiro generalize the full Kostan t-T o da ﬂo ws and the k -c hop construction of the integrals in Proposition 3.3 to arbitrary simple Lie algebras, showing that the T o da ﬂows on a generic coadjoint orbit in a simple Lie algebra g are completely integrable. A k ey observ ation in making this extension is that the 1-c hop matrix φ 1 ( X ) can be obtained as the middle ( n − 2) × ( n − 2) blo c k of Ad Γ( X ) ( X ), where Γ( X ) is a sp ecial elemen t of the Borel subgroup of G . This allo ws the authors to use the adjoint action of a Borel subgroup, follo wed by a pro jection onto a subalgebra, to deﬁne the appropriate analog of the 1-c hop matrix. Finally , we note that full Kostant-T oda lattice has a symmetry of order tw o induced by the non trivial automorphism of the Dynkin diagram of the Lie algebra sl ( n, C ). In terms of the matrices in  + B − , the in v olution is reﬂection along the anti-diagonal. It is sho wn in [67] that this inv olution preserv es all the k -chop in tegrals and thus deﬁnes an in volution on eac h lev el set of the constan ts of motion. In the ﬂag manifold, the symmetry in terc hanges the t wo ﬁxed points of the torus action that corresp ond to antipo dal vertices of the momen t p olytop e under the moment map (3.1). Example 3.1. In this example, w e demonstrate the complexity of the Poisson stratiﬁcation of  + B − for n = 3 and n = 4. The table of symplectic leav es of all dimensions has b een calculated in notes by Stephanie Singer, a co-author of [26], as given b elow. On the leav es of low er dimensions, the k -chop in tegrals are dep endent. When n = 3,  + B − =      f 1 1 0 g 1 f 2 1 h g 2 f 3   : 3 P i =1 f i = 0    . Its symplectic leav es are listed in the following table. The Casimirs are constants of motion that generate trivial Hamiltonian ﬂo ws. The v alue of each Casimir is ﬁxed on a giv en symplectic leaf. THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 25 Closed Conditions Op en Conditions Casimirs Dimension — h 6 = 0 f 2 − g 1 g 2 h 4 h = 0 g 1 g 2 6 = 0 — 4 h, g 1 = 0 g 2 6 = 0 f 1 2 h, g 2 = 0 g 1 6 = 0 f 3 2 h, g 1 , g 2 = 0 — f 1 , f 2 , ( f 3 ) 0 On each 4-dimensional leaf, the functions 1 2 tr( X 2 ) and 1 3 tr( X 3 ) provide a complete family of integrals (Hamiltonians) for the T o da hierarch y . When n = 4, the symplectic stratiﬁcation is already muc h more complicated. Here,  + B − =            f 1 1 0 0 g 1 f 2 1 0 h 1 g 2 f 3 1 k h 2 g 3 f 4     : 4 P i =1 f i = 0        . The table of symplectic lea ves is as follo ws: Closed Conditions Op en Conditions Casimirs Dimension – k ( k g 2 − h 1 h 2 ) 6 = 0 I 11 = − f 2 − f 3 + g 1 h 2 + g 3 h 1 k 8 k g 2 − h 1 h 2 = 0 k 6 = 0 C 1 = f 2 − g 1 h 2 k , C 2 = f 3 − g 3 h 1 k 6 k = 0 h 1 h 2 ( g 1 h 2 + g 3 h 1 ) 6 = 0 — 8 k , h 1 = 0 h 2 g 1 6 = 0 C 3 = f 3 − g 2 g 3 h 2 6 k , h 2 = 0 h 1 g 3 6 = 0 C 4 = f 2 − g 1 g 2 h 1 6 k , g 1 h 2 + g 3 h 1 = 0 h 1 h 2 6 = 0 C 5 = f 1 + f 3 − g 2 g 3 h 2 6 k , h 1 , h 2 = 0 g 1 g 2 g 3 6 = 0 — 6 k , h 1 , g 1 = 0 h 2 6 = 0 f 1 , C 6 = f 3 − g 2 g 3 h 2 4 k , h 1 , g 1 , h 2 = 0 g 2 g 3 6 = 0 f 1 4 k , h 2 , g 3 = 0 h 1 6 = 0 f 4 , C 7 = f 2 − g 1 g 2 h 1 4 k , h 2 , g 3 , h 1 = 0 g 1 g 2 6 = 0 f 4 4 k , h 1 , h 2 , g 2 = 0 g 1 g 3 6 = 0 f 1 + f 2 (= − ( f 3 + f 4 )) 4 k , h 1 , h 2 , g 1 , g 2 = 0 g 3 6 = 0 f 1 , f 2 2 k , h 1 , h 2 , g 2 , g 3 = 0 g 1 6 = 0 f 3 , f 4 2 k , h 1 , h 2 , g 1 , g 3 = 0 g 2 6 = 0 f 1 , f 4 2 k , h 1 , h 2 , g 1 , g 2 , g 3 = 0 — f 1 , f 2 , f 3 , ( f 4 ) 0 On the maximal leav es, of dimension 8, the functions 1 k tr X k for k = 2 , 3 , 4 pro vide three constants of motion. One 1-c hop in tegral is needed to complete the family . 26 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 11 20 21 10 1210 1021 0121 2101 1201 1012 01 12 11 02 021 1 201 1 Figure 3.2. The momen t polytop e ∆ 4 for S L (4 , C ) /P . The v ertices represen t the w eights L i − L j whic h are expressed by i 1 L 1 + i 2 L 2 + i 3 L 3 + i 4 L 4 using L 1 + · · · + L 4 = 0, e.g. 2110 means L 1 − L 4 . 3.4. Nongeneric ﬂo ws in the full Kostan t-T o da lattice. When eigen v alues of the initial matrix in  + B − coincide, the torus embedding (3.22) is not deﬁned since an y matrix in  + B − has one Jordan block for eac h eigen v alue. In the most degenerate case of non-distinct eigen v alues, that is, when all eigenv alues are zero, the isosp ectral set can b e embedded into the ﬂag manifold by the companion em b edding (3.21). Under this em b edding, the 0-chop in tegrals generate the action of the exp onential of an ab elian nilp otent algebra [65]. The 1-c hop in tegrals are again deﬁned only in terms of the partial ﬂag manifold S L ( n, C ) /P . Fixing the v alues of each 1-c hop in tegral produces a v ariety in the ﬂag manifold. The common intersection of all these v arieties turns out to be in v arian t under the action of the diagonal torus and has a simple description in terms of the moment polytop e [65]. [70] considers lev el sets where the eigen v alues of eac h φ k ( X ) are distinct but one or more eigen- v alues of φ j ( X ) and φ j +1 ( X ) coincide for one or more v alues of j . In this situation, the torus orbits generated by the k -chop integrals in the pro duct (3.23)degenerate into unions of nongeneric orbits. The nature of this splitting can b e seen in terms of the momen t polytop es of the partial ﬂag manifolds in (3.23). Recall the deﬁnition of the moment map µ in (3.1). Here G is S L ( n, C ) and V is the adjoin t represen tation. V ma y b e realized as the subspace of C n ⊗ ∧ n − 1 C n with P e i ⊗ e ∗ i = 0, where { e i } is the standard basis of C , and e ∗ i = ( − 1) i +1 e 1 ∧ . . . ∧ e i − 1 ∧ e i +1 ∧ . . . ∧ e n . The partial ﬂag manifold S L ( n, C ) /P is the orbit of G through [ e 1 ⊗ e ∗ n ] in P ( V ). The w eight of e i ⊗ e ∗ j is L i − L j , where L k is the linear function in H ∗ that sends an elemen t of H to its k th diagonal en try . The w eigh ts L i − L j with i 6 = j are the v ertices of the w eight p olytop e of V , which w e denote by 4 n . These v ertices are the images under the momen t map of the ﬁxed points of the complex diagonal torus. The image of the closure of a torus orbit under momen t map is the con v ex h ull of the weigh ts corresp onding to the ﬁxed points of the torus in the closure of the orbit. The real dimension of the image is equal to the complex dimension of the orbit [5]. Figure 3.2 sho ws the example of the momen t polytop e ∆ 4 . An elemen t g B in S L ( n, C ) /P represents the partial ﬂag V 1 ⊂ V n − 1 ⊂ C n where V 1 is the span of the ﬁrst column of g and V n − 1 is the span of the ﬁrst n − 1 columns. There are tw o natural pro jections from S L ( n, C ) /P to the pro jective space CP n − 1 and its dual ( CP n − 1 ) ∗ that send a partial ﬂag to the line V 1 and to the h yp erplane V n − 1 , resp ectiv ely . Let π i and π ∗ i b e pro jective co ordinates on CP n − 1 and ( CP n − 1 ) ∗ . The coordinates π L i − L j that come from the em b edding of S L ( n, C ) /P into P ( V ) b y the moment map (3.1) are pro jectively equal to the products π i π ∗ j for i 6 = j : [ π L i − L j ] i 6 = j = [ π i π ∗ j ] i 6 = j . THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 27 11 20 21 10 1210 1021 0121 2101 1201 1012 01 12 11 02 021 1 201 1 1201 2101 201 1 1021 0121 021 1 Figure 3.3. The polytop es ∆ 4 ( \ k ) and ∆ 4 ( \ k ∗ ), obtained b y splitting ∆ 4 along an in terior hexagon. Eac h is missing the vertices of one triangular face. A t each ﬁxed p oint of the diagonal torus in S L ( n, C ) /P , exactly one π i and one π ∗ j do es not v anish. Those where π k 6 = 0 corresp ond to the vertices L k − L i with i 6 = k , whose conv ex hull is an ( n − 2)-dimensional face of 4 n , whic h w e denote as 4 n ( k ). The ﬁxed p oints where π ∗ k 6 = 0 correspond to the vertices L i − L k of the antipo dal face, 4 n ( k ∗ ). The p olytop e of an ( n − 1)-dimensional torus orbit where π k or π ∗ k is the only v anishing coordinate is the conv ex h ull of the v ertices remaining after the vertices of the face 4 n ( k ), respectively 4 n ( k ∗ ) are remov ed. These p olytop es are denoted 4 n ( \ k ) and 4 n ( \ k ∗ ), resp ectiv ely . They are congruen t polytop es, obtained by splitting 4 n along the h yp erplane through the vertices L i − L j with i, j 6 = k . The conv ex h ull of these vertices is an ( n − 2)-dimensional p olytop e in the interior of 4 n , which w e denote as 4 n ( \ k \ k ∗ ). W e will refer to the pair 4 n ( \ k ) and 4 n ( \ k ∗ ) as a split p olytop e . In Figure 3.3, w e illustrate the example of the split p olytop e ∆ 4 ( \ k ) and ∆ 4 ( \ k ∗ ) [70]. When tw o or more such splittings occur simultaneously , the collection of resulting p olytop es will also b e called a split p olytop e. Prop osition 3.4. [70] L et F b e a variety in S L ( n, C ) /P deﬁne d by ﬁxing the values of the 1-chop inte gr als I r 1 , including the Casimir, wher e the values ar e chosen so that exactly one eigenvalue, say λ i 0 , of X is also an eigenvalue of φ 1 ( X ) . Then F is the union of the closur es of two nongeneric torus orbits, O i and O i ∗ , on which π i , r esp e ctively π ∗ i , is the only c o or dinate that vanishes. The images of their closur es under the moment map, 4 n ( \ i ) and 4 n ( \ i ∗ ) , ar e obtaine d by splitting 4 n along the interior ( n − 2) -dimensional fac e 4 n ( \ i \ i ∗ ) . When exactly p eigenvalues of X ar e also eigenvalues of φ 1 ( X ) ( p ≤ n − 2 ), then F is the union of the closur es of 2 p nongeneric ( n − 1) - dimensional orbits whose images under the moment map ar e the p olytop es obtaine d by splitting 4 n simultane ously along p interior fac es 4 n ( \ j \ j ∗ ) . This result extends to the k -chop ﬂo ws as follows: Prop osition 3.5. [70] If p eigenvalues of φ k ( X ) and φ k − 1 ( X ) c oincide, then the generic orbit of the diagonal torus that gener ates the ( k − 1) -chop ﬂows in the c omp onent S L ( n − 2( k − 1) , C ) /P of (3.23) b e c omes a union of 2 p nongeneric orbits. Sinc e the moment map on the pr o duct (3.23) is the pr o duct of the c omp onent moment maps, the moment map on the pr o duct of p artial ﬂag manifolds takes a level set in (3.23) to a pr o duct of ful l and/or split p olytop es, dep ending on wher e the c oincidenc es of eigenvalues o c cur. When a lev el set of the constan ts of motion is split into t wo or more nongeneric torus orbits, there are separatrices in the T o da ﬂo ws that generate the torus action. The faces along whic h the p olytop e is split are the images under the moment map of low er-dimensional torus orbits (the separatrices) that form the in terface betw een the nongeneric orbits of maximum dimension. The ﬂo w through an 28 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 initial condition in one maximal orbit is conﬁned to that orbit. It is separated from the ﬂows in the complemen tary nongeneric orbits b y the separatrices. [68] determines the monodromy around these singular level sets in the ﬁb er bundle of lev el sets where the sp ectrum of the initial matrix is ﬁxed with distinct eigenv alues and the remaining constant of motion I (equiv alen t to the determinant of the 1-c hop matrix) is allow ed to v ary . The ﬂo w generated b y I pro duces a C ∗ -bundle with singular ﬁb ers ov er the v alues of I . The singularities o ccur b oth at v alues of I where an eigenv alue of the 1-c hop matrix coincides with an eigen v alue of the original matrix and at v alues of I where the t wo eigen v alues of the 1-chop matrix coincide. In a neigh b orho o d of a singular ﬁber of the ﬁrst kind, the mono drom y is c haracterized by a single t wist of the noncompact cycle around the cylinder C ∗ . Near a singular ﬁb er of the second kind, the mono dromy pro duces tw o twists of the noncompact cycle. This double t wist is seen in the simplest case when n = 2 near the ﬁb er where the tw o eigen v alues of the original matrix coincide; it as describ ed in detail in [66]. When eigen v alues of φ 1 ( X ) coincide, the torus em b edding (3.22) generalizes to the Jordan em b ed- ding (3.15), under which the 1-c hop ﬂows generate the action of the group A J in (2.20), a pro duct of a diagonal torus and a nilpotent group. The general structure of a lev el set of the 1-chop integrals with this type of singularity is not known, in part b ecause the orbit structure of A J in the ﬂag manifold is not understoo d in suﬃcien t detail. When the eigenv alues are distinct, A J is a diagonal torus, and the closures of its orbits are toric v arieties [59]. The structure of torus orbits in ﬂag manifolds is well-understoo d; see for example [5], [30], and [34]. The closures of orbits of A J in the ﬂag manifold are generalizations of toric v arieties, and muc h less is known about them. The ﬁxed p oints of the actions of the groups A J are studied in [69], and the ﬁxed point sets of the torus on the diagonal of A J are characterized in [71]. If A J has r blo cks along the diagonal, where the dimension of the i th blo ck is d i , then the maximal diagonal subgroup of A has gcd( d 1 , . . . , d r ) connected comp onents [75]. The subgroup of A J that ﬁxes all p oints in the ﬂag manifold is the discrete group D consisting of constant multiples of the identit y where the constan ts are the n th ro ots of unity; the group A J /D then acts eﬀectively on the ﬂag manifold. [74] describes the ﬁxed- p oin t set of the unip otent part of A J , giving an explicit wa y to express it in terms of canonical co ordinates in each Bruhat cell. In the case where all eigenv alues coincide, A J is equal to its unip oten t part. [73] sho ws that the action of the group in this case preserv es each Bruhat cell and that its orbits in a giv en cell are c haracterized by the ”gap sequence” of the p erm utation associated to the cell. 4. Other Extensions of the Toda La ttice 4.1. Isosp ectral deformation of a general matrix. In the full Hessen b erg form of the T o da lattice, the matrix is diagonalizable if and only if the eigenv alues are distinct. Kodama and Y e generalize this in [52], where they consider an iso-spectral deformation of an arbitrary diagonalizable matrix L . The ev olution equation is (4.1) d dt L = [ P , L ] ; P is deﬁned b y (4.2) P = Π( L ) = ( L ) > 0 − ( L ) < 0 , where ( L ) > 0( < 0) is the strictly upp er (low er) triangular part of L . [52] establishes the complete in tegrability of (4.1) using in verse scattering, generalizing the metho d used in [47] to solve the full symmetric real T o da lattice. The method yields an explicit solution to the initial-v alue problem. The general context of the ﬂow (4.1) includes as sp e cial cases the T o da lattices on other classical Lie algebras in addition to sl ( n, R ), whic h is most closely asso ciated with T oda’s original system. In this regard, Bogo y a vlensky in [12] form ulated the T o da lattice on the real split semisimple Lie algebras, THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 29 whic h are deﬁned as follows (the form ulation b elow is in the Hessen berg (or Kostan t) form): Let { h α i , e ± α i : i = 1 , . . . , l } b e the Chev alley basis of the algebra g of rank l , i.e. [ h α i , h α j ] = 0 , [ h α i , e ± α j ] = ± C j i e ± α j , [ e α i , e − α j ] = δ ij h α j , where ( C ij ) 1 ≤ i,j ≤ l is the Cartan matrix and C ij = α i ( h α j ). Then the (non-perio dic) T o da lattice asso ciated with the Lie algebra g is deﬁned b y the Lax equation (4.3) dL dt = [ A, L ] , where L is a Jacobi elemen t of g and A is the N − -pro jection of L , L ( t ) = l P i =1 f i ( t ) h α i + l P i =1 ( g i ( t ) e − α i + e α i ) A ( t ) = − Π N − L ( t ) = − l P i =1 g i ( t ) e − α i . The complete integrabilit y is based on the existence of the Chev alley in v ariants of the algebra, and the geometry of the isosp ectral v ariety has b een discussed in terms of the represen tation theory of Lie groups b y Kostan t in [54] for the cases where g i are real p ositiv e, or complex. The general case for real g i ’s is studied b y Casian and Kodama [15, 16], which extends the results in the sl ( n, R ) T o da lattice in the Hessenberg form (see Section 2.4) to the T o da lattice for any real split semisimple Lie algebra. The Lax equation (4.3) then giv es d f i dt = g i dg i dt = − l P j =1 C ij f j ! g i from whic h the τ -functions are deﬁned as (4.4) f k ( t ) = d dt ln τ k ( t ) , g k ( t ) = g k (0) l Y j =1 ( τ j ( t )) − C kj . In the case of g = sl ( n, R ), those equations are (2.28) and (2.29) (note here that the superdiagonal of L ( t ) is diag( f 1 − f 2 , f 2 − f 3 , . . . , f l − f l +1 ) with n = l + 1). Those extensions hav e b een discussed by man y authors (see for example [36, 60]). One should note that Bogoy avlensky in [12] also formulates those T oda lattices for aﬃne Kac-Mo o dy Lie algebras, and they give the p erio dic T oda lattice. There has b een m uc h imp ortant progress on the p erio dic T o da lattices, but we will not cov er the sub ject in this pap er (see for example [2, 3, 24, 62, 63]). F rom the viewp oint of Lie theory , the underlying structure of the integrable systems is based on the Lie algebra splitting, e.g. sl ( n ) = B − ⊕ so ( n ) (the QR-decomp osition) for the symmetric T o da lattice, and sl ( n ) = B + ⊕ N − (the Gauss decomposition) for the Hessenberg form of T o da lattice. Then one can also consider the follo wing form of the evolution equation, (4.5) d dt L = [ Q k , L ] with Q k = Π g 1 ( L k ) , where g 1 is a subalgebra in the Lie algebra splitting sl ( n ) = g 1 ⊕ g 2 . In this regard, we men tion here the follo wing tw o interesting systems directly connecting to the T o da lattice: 30 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 (a) The Kac-v an Mo erb ek e system [42]: W e tak e g 1 = so (2 n ), and consider the equation for L ∈ so (2 n ) (recall that L is a symmetric matrix for the symmetric T o da lattice) . Since L 2 k − 1 ∈ so (2 n ), the ev en ﬂows are all trivial. Let L b e given b y a tridiagonal form, L =        0 α 1 0 · · · 0 − α 1 0 α 2 · · · 0 . . . . . . . . . · · · · · · 0 0 · · · 0 α 2 n − 1 0 0 · · · − α 2 n − 1 0        ∈ so (2 n, R ) Then the even ﬂows are the Kac-v an Mo erb ek e hierarch y , ∂ L ∂ t 2 j = [Π so ( L 2 j ) , L ] (recall that Π so ( L 2 j ) = Sk ew( L 2 j )), where the ﬁrst mem b er of t 2 -ﬂo w gives ∂ α k ∂ t 2 = α k ( α 2 k − 1 − α 2 k +1 ) , k = 1 , . . . , 2 n − 1 , with α 0 = α 2 n = 0. This system is equiv alent to the symmetric T o da lattice which can b e written as (4.5) for the square L 2 . Note here that L 2 is a symmetric matrix giv en by L 2 = T (1) ⊗  1 0 0 0  + T (2) ⊗  0 0 0 1  , where T ( i ) , for i = 1 , 2, are n × n symmetric tridiagonal matrices given b y T ( i ) =         b ( i ) 1 a ( i ) 1 0 · · · 0 a ( i ) 1 b ( i ) 2 a ( i ) 2 · · · 0 . . . . . . . . . . . . . . . 0 0 · · · b ( i ) n − 1 a ( i ) n − 1 0 0 · · · a ( i ) n − 1 b ( i ) n         , with a (1) k = α 2 k − 1 α 2 k , b (1) k = − α 2 2 k − 2 − α 2 2 k − 1 , a (2) k = α 2 k α 2 k +1 , and b (2) k = − α 2 2 k − 1 − α 2 2 k (see [35]). Then one can sho w that each T ( i ) giv es the symmetric T oda lattice, that is, the Kac-v an Mo erb eke hierarc hy for L 2 matrix splits in to tw o T o da lattices, ∂ T ( i ) ∂ t 2 j = [Π so ( T ( i ) ) j , T ( i ) ] i = 1 , 2 . The equations for T ( i ) are connected b y the Miura-type transformation, with the functions ( a ( i ) k , b ( i ) k ), through the Kac-v an Mo erb eke v ariables α k (see [35]). (b) The Pfaﬀ lattice for a symplectic matrix [1, 48]: The Pfaﬀ lattice is deﬁned in the same form with g 1 = sp (2 n ) and L in the Hessen be rg form with 2 × 2 blo ck structure. In particular, w e consider the case L ∈ sp (2 n ) having the form, L =             0 s 1 b 1 0 0 0 a 1 0 · · · 0 2 0 0 a 1 0 0 s 2 b 2 0 · · · 0 2 . . . . . . . . . . . . 0 2 0 2 · · · 0 s n b n 0             ∈ sp (2 n, R ) , THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 31 where 0 2 is the 2 × 2 zero matrix. The v ariables ( a k , b k ) and s k = ± 1 are those in the indeﬁnite T o da lattice. It should be noted again that the odd members are trivial (since L 2 k − 1 ∈ sp (2 n )), and the ev en members giv e the indeﬁnite T o da lattice hierarch y [49]. Here one should note that L 2 can b e written as L 2 = ˜ L T ⊗  1 0 0 0  + ˜ L ⊗  0 0 0 1  , where ˜ L is giv en b y (2.33). Then one can show that the generator Q 2 j of the Lax equation is giv en b y Q 2 j = Π sp ( L 2 j ) = − ˜ B T j ⊗  1 0 0 0  + ˜ B j ⊗  0 0 0 1  , where ˜ B j = 1 2 [( ˜ L j ) > 0 − ( ˜ L j ) < 0 ]. Then the hierarc h y d dt L = [ Q 2 j , L ] gives the indeﬁnite T o da lattice hierarc hy . 4.2. Gradien t form ulation of T o da ﬂows. In [7], Bloch observ ed that the symmetric tridiagonal T o da equations (2.6) can also b e written in the double-brack et form (4.6) d dt L ( t ) = [ L ( t ) , [ L ( t ) , N ]] , where N is the constan t matrix diag(1 , 2 , ..., n ) and L is as in (2.7). He show ed that this double- brac ket equation is the gradien t ﬂow of the function f ( L ) = tr( LN ) with resp ect to the normal metric on an adjoint orbit of S O ( n ). The normal metric is deﬁned as follo ws: Let κ ( , ) = −h , i b e the Killing form of a semisimple Lie algebra g , and decomp ose g orthogonally relative to h , i into g = g L ⊕ g L where g L is the cen tralizer of L and g L = Im ad( L ). F or X ∈ g , denote b y X L the pro jection of X on to g L . Then giv en t w o tangen t v ectors to the orbit at L , [ L, X ] and [ L, Y ], the normal metric is deﬁned b y h [ L, X ] , [ L, Y ] i N = h X L , Y L i . Then the righ t hand side of (4.6) can b e written as grad H = [ L, [ L, N ]] for the Hamiltonian function H ( L ) = κ ( L, N ) (Prop osition 1.4 in [9]). Thus the T o da lattice (2.6) is both Hamiltonian and a gradient ﬂo w on the isosp ectral set. Bro c kett shows in [13] that any symmetric matrix L (0) can b e diagonalized by the ﬂow (4.6), and the ﬂo w can b e used to solve v arious com binatorial optimization problems such an linear programming problems (see [37] for the connections of the T o da lattice with several optimization problems). The ﬂow (4.6) is extended in [8] and [9] to show that the generalized tridiagonal symmetric T o da lattice can also be expressed as a gradien t ﬂo w. In Section 4.1, w e giv e the equations of the generalized tridiagonal T o da lattice in the Hessenberg form on a real split semisimple Lie algebra. The symmetric version of this is as follows (see [9]). Let g b e a complex semisimple Lie algebra of rank l with normal real form g n . Cho ose a Chev alley basis { h α k , e ± α k : k = 1 , . . . , l } as in Section 4.1. The generalized tridiagonal symmetric T o da lattice is deﬁned by the Lax equation d dt L ( t ) = [ A ( t ) , L ( t )] , where L ( t ) = l P k =1 b k ( t ) h α k + l P k =1 a k ( t )( e α k + e − α k ) A ( t ) = l P k =1 a k ( t ) ( e α k − e − α k ) . This ﬂo w deﬁnes a completely in tegrable Hamiltonian system on the coadjoint orbit of the low er Borel subalgebra of g n through l P k =1 ( e α k − e − α k ). The Hamiltonian is H ( L ) = 1 2 K ( L, L ), where K is the Killing form. 32 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 [9] sho ws that (4.2) is a gradient ﬂo w with respect to the normal metric on the orbit. The gradient form ulation in [9] is giv en in the con text of the compact form g u of g (see also the survey in [11]). Their k ey result is the following. Prop osition 4.1. The gr adient ve ctor ﬁeld of the function f ( L ) = K ( LN ) on the adjoint orbit in g u c ontaining the initial c ondition L 0 , with r esp e ct to the normal metric, is (4.7) d dt L ( t ) = [ L ( t ) , [ L ( t ) , N ]] . No w let H u b e a maximal ab elian subalgebra of g u , and take H = H u ⊕ iH u as the Cartan subalgebra of g . Cho ose a Chev alley basis for g as ab ov e. Blo ch, Bro ck ett, and Ratiu [9] show the follo wing. Theorem 4.1. L et N b e i times the sum of the simple c oweights of g , and let L ( t ) = l P k =1 i b k ( t ) h α k + l P k =1 i a k ( t )( e α k + e − α k ) . Then the gr adient ve ctor ﬁeld (4.7) gives the ﬂow of the gener alize d tridiagonal symmetric T o da lattic e on the adjoint orbit in g u c ontaining the initial c ondition L 0 . Explicitly, N = l P k =1 i x k h α k , wher e ( x 1 , ..., x l ) is the unique solution of the system l P k =1 x k α p ( h α k ) = − 1 , p = 1 , ..., l . A list of the co eﬃcients ( x 1 , ..., x l ) for all the semi-simple Lie algebras is given on p. 62 of [9]. Prop osition (4.1) and Theorem (4.1) are extended in [23] to the generalized full symmetric T o da lattice and in [10, 11] to the generalized signed T oda lattice to show that these extensions of the T o da ﬂows are also gradien t ﬂows. 5. Connections with the KP equa tion Here we giv e a brief review of the pap er [6] whose main result is to show that the τ -functions of the T o da hierarc hy (2.44) with a symmetric tridiagonal matrix pro vide a new class of solutions of the Kadom tsev-Petviash vili (KP) equation. W e also provide a geometric description of the τ -functions in terms of the Grassmann manifolds (see [46]). 5.1. The τ -functions for the symmetric T o da lattice hierarch y. W e return to the T o da lattice hierarc hy (2.44) with symmetric tridiagonal matrix. The solution L ( t ) can b e explicitly expressed in terms of the τ -functions (2.13): Let us summarize the pro cess of solution metho d based on the Gram-Sc hmidt orthogonalization. First w e consider g ( t ) = exp  1 2 θ ( L (0) , t )  , where θ ( λ, t ) := n − 1 P k =1 t k λ k with the k -th ﬂow parameter t k of the T o da hierarch y , i.e. ∂ L ∂ t k = [ B k , L ] with B k = 1 2 Sk ew( L k ) . THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 33 (Note here that w e rescale the time t k → t k / 2.) Then deﬁne the matrix M ( t ) := g T ( t ) g ( t ) = e θ ( L (0) , t ) = Φ(0) e θ (Λ , t ) Φ T (0) =  h φ 0 i φ 0 j e θ ( λ, t ) i  1 ≤ i,j ≤ n where Φ(0) = ( φ 0 i ( λ j )) 1 ≤ i,j ≤ n is the eigenmatrix of L (0), i.e. L (0)Φ(0) = Φ(0)Λ, and Φ(0) ∈ S O ( n ). Since L (0) is a tridiagonal matrix, the en tries m i,j ( t ) := h φ 0 i φ 0 j e θ ( λ, t ) i can be written in terms of the momen t by the Gram-Sc hmidt orthogonalization pro cess (see [47] for the details), m i,j ( t ) = h λ i + j − 2 ρ ( λ ) e θ ( λ, t ) i = n P k =1 λ i + j − 2 k ρ k e θ k ( t ) , where ρ ( λ ) = φ 0 1 ( λ ) 2 with ρ k = ρ ( λ k ), and θ k ( t ) = θ ( λ k , t ). In particular, we ha v e (5.1) τ 1 ( t ) = h ρ ( λ ) e θ ( λ, t ) i = n P k =1 ρ k e θ k ( t ) . Then the τ -functions are giv en b y the W ronskian of the set of functions of τ 1 ( t ) and its x -deriv atives, τ k ( t ) = W r ( τ 1 ( t ) , τ 0 1 ( t ) , . . . , τ ( k − 1) 1 ) for k = 1 , 2 , . . . , n − 1 . Using the Binet-Cauc hy theorem, one can write τ k in the form (5.2) τ k ( t ) = P 1 ≤ i 1 < ··· 0 . As the simplest case, let us consider the sl (2 , R ) T o da lattice: W e hav e one τ -function, τ 1 ( t ) = s 1 ρ 1 e λ 1 t + s 2 ρ 2 e λ 2 t . If s 1 s 2 < 0, τ 1 ( t ) has zero at a time t = 1 λ 2 − λ 1 ln( ρ 1 ρ 2 ), that is, we hav e a blo w-up in the solution. The image of the moment map µ ( τ 1 ) is given by a line segment whose end points corresp ond to the weigh ts L 1 and L 2 = − L 1 . Although the dynamics are so diﬀerent in the cases s 1 s 2 > 0 and s 1 s 2 < 0, the moment polytop e (a line segment) is independent of the signs of the s i ’s. Notice that s 1 s 2 = sgn( g 1 ), and in general, if sgn( g k ) < 0 for some k , then the solution blows up sometime in R . In order to ﬁnd the general pattern of the sign changes in ( g 1 ( t ) , . . . , g n − 1 ( t )) of the matrix X in (2.22), we ﬁrst recall that the isospectral v ariety is characterized by the moment p olytop e M  whose vertices are given b y the orbit of W eyl group action. Here the set of signs  = (  1 , . . . ,  n − 1 ) is deﬁned b y the signs of g i for t → −∞ . F rom the ordering λ 1 < · · · < λ n , we ﬁrst see that τ k ( t ) ≈ s 1 · · · s k K (1 , . . . , k ) exp(( λ 1 + · · · + λ k ) t ). Then from the deﬁnition of g k ( t ) in (2.28), i.e. g k = τ k − 1 τ k +1 /τ 2 k , the sign of g k ( t ) for t → −∞ is given b y  k = sgn( g k ) = s k s k +1 for k = 1 , . . . , n − 1 . Then from the moment map (5.17), one notes that the momen t p olytop e given as the image of the momen t map µ ( M  ) in (6.4) is indep enden t of the sign set  . How ev er the dynamics of the T o da lattice with a diﬀeren t  is quite diﬀeren t, and the solution with at least one  k < 0 has a blow-up at some t ∈ R . W e no w consider each edge of the p olytop e which corresponds to an sl (2 , R ) indeﬁnite T oda lattice, that is, where g j 6 = 0 for only one j . This edge can be also expressed by a simple reﬂection r j ∈ W . Since the simple reﬂection r j exc hanges s j and s j +1 , w e hav e an action of r j on all the signs  k , r j :  k →  0 k ,  0 k = r j (  k ) =     k  k − 1 if j = k − 1  k  k +1 if j = k + 1  k if j = k , or | j − k | > 1 whic h can b e also shown directly from the form of τ k ( t ) in (6.7). This formula can be extended to the indeﬁnite T oda lattice on an y real split semisimple Lie algebras, and we hav e (see (4.4) and Prop osition 3.16 in [15]): 46 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 Prop osition 6.1. L et  j = sgn( g j ) for j = 1 , . . . , n − 1 . Then the Weyl gr oup action on the signs is given by r j :  k 7→  k  − C kj j , wher e ( C ij ) 1 ≤ i,j ≤ n − 1 is the Cartan matrix of sl ( n, R ) . With this W -action on the signs  = (  1 , . . . ,  n − 1 ) with  k = sgn( g k ) at each vertex of the p olytop e, we now deﬁne the relation b etw een the vertices lab eled by w and w 0 = wr i as follo ws: Notice that if  i = +, then (  1 , · · · ,  n − 1 ) remains the same under r i -action. Then w e write w = ⇒ w 0 with w 0 = w r i . No w the follo wing deﬁnition giv es the n um b er of blo w-ups in the T o da orbit from the top v ertex e to the v ertex lab eled by w ∈ W : Cho ose a reduced expression w = r j 1 · · · r j k . Then consider the sequence of the signs at the orbit given by w -action,  → r j 1  → r j 2 r j 1  → · · · → w − 1  . W e then deﬁne the function η ( w ,  ) as the n umber of → which are not of the form ⇒ . The n um b er η ( w ∗ ,  ) for the longest elemen t w ∗ giv es the total num b er of blow-ups along the T o da ﬂow in the p olytop e of M  . Whenever  = ( − , . . . , − ), we just denote η ( w ,  ) = η ( w ). This n umber η ( w ,  ) do es not dep end on the choice of the reduced expression of w (see Corollary 5.2 in [18]). Hence the num b er of blow-up p oin ts along the tra jectories in the edges of the polytop e is independent of the tra jectory parametrized by the reduced expression. In Figure 2.3, we illustrate the num bers η ( w ,  ) for the sl (3 , R ) indeﬁnite T oda lattice. F or example, on M −− , w e ha ve η ( e ) = 0 , η ( r 1 ) = η ( r 2 ) = η ( r 1 r 2 ) = η ( r 2 r 1 ) = 1 and η ( r 1 r 2 r 1 ) = 2, i.e the total num b er of blo w-ups is 2. W e also illustrate this for the sl (4 , R ) T o da lattice in Figure 6.3. Along the path shown in this Figure, w e ha ve η ( e ) = 0 , η ([2]) = η ([21]) = η ([213]) = 1 , η ([2132]) = 2 , η ([21323]) = 3 and η ( w ∗ ) = 4, where [ ij · · · k ] = r i r j · · · r k , and note [21323] = [12312]. In general, the total n umber of blow-ups η ( w ∗ ,  ) dep ends only the initial signs  = (  1 , . . . ,  n − 1 ) with  i = sgn( g i ( t )) for t → −∞ , which is given by  i = s i s i +1 . Then in the case of sl ( n, R ) indeﬁnite T o da lattice, the n umber η ( w ∗ ,  ) = m ( n − m ) where m is the total num b er of negative s i ’s (Proposition 3.3 in [51]). In particular, the maxim um n umber of blo w-ups o ccurs the case with  = ( − , . . . , − ), and it is giv en by [( n + 1) / 2]( n − [( n + 1) / 2]). Those n umbers η ( w ∗ ,  ) are related to the p olynomials app earing in F q p oin ts on certain compact groups deﬁned in (6.6). W e now in tro duce p olynomials in terms of the num b ers η ( w ,  ), whic h play a k ey role for counting the num ber of blow-ups and give a surprising connection to the rational cohomology of the maximal compact subgroup S O ( n ) (Deﬁnition 3.1 in [18]). Deﬁnition 6.3. W e deﬁne a monic p olynomial asso ciated to the p olytop e M  , p ( q ,  ) = ( − 1) l ( w ∗ ) P w ∈ W ( − 1) l ( w ) q η ( w, ) , where l ( w ) indicates the length of w . Notice that the degree of p ( q ,  ), denoted b y deg( p ( q ,  )), is the total num b er of blo w-ups, i.e. η ( w ∗ ,  ) = deg( p ( q ,  )). F or the case  = ( − , . . . , − ), we simply denote it b y p ( q ). Example 6.4. In the case of the sl (2 , R ) T o da lattice, (a) for  = (+), we ha v e e ⇒ s 1 whic h gives p ( q, +) = 0, (b) for  = ( − ), we ha v e a blow-up betw een e and s 1 , hence p ( q , − ) = q − 1. Recall from the previous section that the p olynomial p ( q ) = p ( q , − ) app ears in | S O (2 , F q ) | = q − 1. In the case of the sl (3 , R ) T o da lattice, from Figure 2.3, (a) for all the cases of  = (  1 ,  2 ) except ( − , − ), w e hav e p ( q ,  ) = 0. THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 47 1 1 1 2 2 4 3 3 e 3 [123] [1232] [1231] [12312] [12321] w =[123121] * Figure 6.3. The momen t polytop e M −−− for the sl (4 , R ) indeﬁnite T oda lattice. The divisors deﬁned b y the set of zero p oints for the τ -functions are sho wn b y the dotted curv e for { τ 1 = 0 } , by the ligh t color one for { τ 2 = 0 } and the dark one for { τ 3 = 0 } . The double circles indicate the divisors with { τ i = 0 } ∩ { τ j = 0 } whic h are all connected at the center of the p olytop e corresp onding to the point with { τ 1 = τ 2 = τ 3 = 0 } . The num bers in the p olytop e indicate the n umber of blo w-ups along the ﬂo w. An example of a path from the top v ertex e to the bottom v ertex w ∗ , the longest elemen t of S 4 , is sho wn by directed edges. (b) for  = ( − , − ), we ha v e p ( q ) = q 2 − 1. Note again that the p olynomial p ( q ) app ears in | S O (3 , F q ) | = q ( q 2 − 1). In the case of sl (4 , R ), we ha v e, from Figure 6.3, (a) for all  = (  1 ,  2 ,  3 ) except ( − , − , − ), p ( q ,  ) = 0. (b) for  = ( − , − , − ), p ( q ) = q 4 − 2 q 2 + 1 = ( q 2 − 1) 2 . Again note that | S O (4 , F q ) | = q 2 ( q 2 − 1) 2 . Casian and Ko dama then pro ve that the polynomial p ( q ) for M  with  = ( − , . . . , − ) in Deﬁnition 6.3 agrees with the p olynomial p ( q ) in | K ( F q ) | in (6.6) where K is the maximal compact subgroup of real split semisimple Lie group G for the T o da lattice (Theorem 6.5 in [18]). Th us the polynomial p ( q ) con tains all the information on the F q p oin ts on the compact subgroup K of G , which is also related to the rational cohomology , i.e. H ∗ ( K, Q ) = H ∗ ( G/B , Q ) (see (6.5)). No w recall that the in tegral cohomology of the real ﬂag v ariet y G/B is obtained b y the incidence graph G G/B in Deﬁnition 6.1. In [18], Casian and Kodama sho w that the graph G G/B can b e obtained from the blo w-ups of the T o da ﬂow. They deﬁne a graph G  asso ciated to the blow-ups as follo ws: Deﬁnition 6.5. The graph G  consists of vertices labeled b y the elements of the W eyl group W and orien ted edges ⇒ . The edges are deﬁned as follows: w 1 ⇒ w 2 iﬀ          (a) w 1 ≤ w 2 (Bruhat order) (b) l ( w 1 ) = l ( w 2 ) + 1 (c) η ( w 1 ,  ) = η ( w 2 ,  ) (d) w − 1 1  = w − 1 2  When  = ( − , . . . , − ), we simply denote G = G  . Then they prov e that G  with  = ( − , . . . , − ) is equiv alent to G G/B (Theorem 3.5 in [18] whic h is the main theorem in the pap er). F or example, the graph G asso ciated with Figure 6.3 agrees with 48 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 the incidence graph G G/B giv en in Figure 6.2. The pro of of the equiv alence G G/B = G contains sev eral technical steps, whic h are b eyond the scope of this review. References [1] M. Adler and P . van Moerb eke, T o da versus Pfaﬀ lattice and related p olynomials, Duke Math. Journal, 112:1–58 (2002) [2] M. Adler and P . v an Moerb eke, Completely in tegrable systems, Euclidian Lie algebras and curv es, Adv. Math. 38: 267–317 (1980) [3] M. Adler and P . v an Mo erb eke, Linearization of Hamiltonian systems, Jacobi v arieties and representation theory , Adv. Math. 38: 318–379 (1980) [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Graduate T exts in Mathematics 60, Springer-V erlag, 1989 [5] M. B. Atiy ah, Conv exity and Commuting Hamiltonians, Bull. London Math. So c. 14:1–15 (1982) [6] G. Biondini and Y. Ko dama, On a family of solutions of the Kadomtsev-P etviashvili equation which also satisfy the T o da lattice hierarch y , J. Phys. A: Math. Gen. 36: 10519–10536 (2003) [7] A. M. Bloch, Steep est descent, linear programming and Hamiltonian ﬂo ws, Contemp. Math. AMS 114:77–88 (1990) [8] A. M. Blo ch, R. W. Bro ck ett, T. S. Ratiu, A new formulation of the generalized T o da lattice equations and their ﬁxed point analysis via the momentum map, Bull. Amer. Math. So c. 23(2): 477–485 (1990) [9] A. M. Blo ch, R. W. Brock ett, T. Ratiu, Completely in tegrable gradien t ﬂows, Comm. Math. Phys. 147: 57–74 (1992) [10] A. M. Blo ch and M. Gekhtman, Hamiltonian and gradien t structures in the T o da ﬂows, J. Geom. Phys. 27: 230–248 (1998) [11] A. M. Blo ch and M. Gekhtman, Lie algebraic asp ects of the ﬁnite nonp erio dic T oda ﬂows, J. Comp. Appl. Math. 202: 3–25 (2007) [12] O. I. Bogoy a vlensky , On p erturbations of the p erio dic T o da lattice, Comm. Math. Phys. 51: 201–209 (1976) [13] R. W. Bro ck ett, Dynamical systems that sort lists and solve linear programming problems, Pr o c. 27th IEEE Confer ence on De cision and Control , Austin, TX, 779–803 (1988) (see also, Linear Algebra Appl. 146: 79–91 (1991)) [14] R. Carter, Simple Gr oups of Lie typ e , (Wiley Classical Library Edition, London, New Y ork, Sidney , T oron to, 1989) [15] L. Casian and Y. Ko dama, T o da lattice and toric v arieties for real split semisimple Lie algebras, Paciﬁc J. Math. 207: 77–123 (2002) [16] L. Casian and Y. Ko dama, Blow-ups of the T o da lattices and their intersections with the Bruhat cells, Comtem- porary Math. 301: 283-310 (2002) [17] L. Casian and Y. Ko dama, Twisted T omei manifolds and the T o da lattices, Contemp. Math. 309: 1–19 (2002) [18] L. Casian and Y. Kodama, T o da lattice, cohomology of compact Lie groups and ﬁnite Chev alley groups, Inv ent. Math. 165: 163–208 (2006) [19] L. Casian and Y. Ko dama, Singular structure of T oda lattices and cohomology of certain compact Lie groups, J. Comp. Appl. Math. 202: 56–79 (2007) [20] L. Casian and R. Stanton, Sch ubert cells and representation theory , Inv ent. Math. 137: 461–539 (1999) [21] S. Chakrav arty and Y. Kodama, Classiﬁcation of the line-soliton solutions of KPII, (arXiv: nlinSI/0710.1456) [22] P . Deift, L. C. Li, T. Nanda, and C. T omei, The T o da ﬂow on a generic orbit is in tegrable, CP AM 39: 183–232 (1986) [23] F. de Mari, M. Pedroni, T o da ﬂows and real Hessenberg manifolds, J. Geom. Anal. 9(4):607–625 (1999) [24] B. A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Sur. 36, No.2, 11–92 (1981) [25] N. M. Ercolani and H. Flasc hk a and L. Haine, Painlev e Balances and Dressing T ransformations, In: Painlev e T ranscendents, NA TO ASI series, Series B, Physics 278 (1991) [26] N. Ercolani, H. Flasc hk a, and S. Singer, The geometry of the full Kostant-T o da lattice In: Integrable Systems, V ol. 115 of Progress in Mathematics 181-226, Birkh¨ auser (1993) [27] H. Flaschk a, The T oda lattice. I. Existence of integrals, 1 Phys. Rev. B 9(4): 1924–1925 (1974) [28] H. Flaschk a, On the T oda lattice. I I. Prog. Theor. Phys. 51(3): 703–716 (1974) [29] H. Flaschk a, Integrable systems and torus actions, Lecture notes, The Universit y of Arizona [30] H. Flaschk a and L. Haine, T orus orbits in G/P , Paciﬁc J. Math. 149(2): 251–292 (1991) [31] H. Flaschk a and L. Haine, V arietes de drap eaux et reseaux de T o da, Math. Z. 208: 545-556 (1991) 1 The title reads II, apparently a misprint. THE FINITE NON-PERIODIC TODA LA TTICE: A GEOMETRIC AND TOPOLOGICAL VIEWPOINT 49 [32] N. C. F reeman and J. J. C. Nimmo, Soliton-solutions of the Korteweg-deV ries and Kadomtsev-P etviashvili equations: the W ronskian technique, Phys. Lett. 95A:1–3 (1983) [33] M. Gekh tman and M. Shapiro, Noncomm utativ e and comm utative integrabilit y of generic T o da ﬂo ws in simple Lie algebras, Comm. Pure & Appl. Math. 52: 53–84 (1999) [34] I. M. Gelfand and V. V. Serganov a, Combinatorial geometries and torus strata on homogeneous compact mani- folds, Usp. Mat. Nauk. 42(2):107–134 (1987) [35] F. Gesztesy , H. Holden, B. Simon and Z. Zhao, On the T o da lattice and Kac-v an Mo erbeke systems, T rans. AMS, 339: 849–868 [36] M. A. Guest, Harmonic Maps, L oop Gr oups, and Inte grable Systems , London Mathematical So ciety Studen t T exts 38 (Cambridge Universit y Press 1997) [37] U. Helmke and J. B. Mo ore, Optimization and Dynamic al Systems , (Splinger-V erlag, London, 1994) [38] M. Henon, Integrals of the T o da lattice, Phys. Rev. B 9:1921–1923 (1974) [39] R. Hirota, The Dir e ct Metho d in Soliton The ory , (Cambridge Universit y Press, Cambridge, 2004) [40] K. Ikeda, Compactiﬁcations of the iso level sets of the Hessen berg matrices and the full Kostant-T o da lattice, Proc. Japan Acad. A 82:93–96 (2006) [41] E. Infeld and G. Rowlands, Nonline ar waves, solitons and chaos , (Cambridge Univ ersity Press, Cam bridge, 2000) [42] M. Kac and P . v an Mo erb eke, On an explicitly soluble system of nnlinear diﬀerential equations related to certain T o da lattices, Adv. Math. 16: 160–169 (1975) [43] B. B. Kadomtsev and V. I. P etviashvili, On the stability of solitary w aves in w eakly disp ersing media, Sov. Ph ys. Doklady, 15: 539–541 (1970) [44] F. C. Kirw an, Cohomolo gy of quotients in symple ctic and algebr aic ge ometry Math. Notes 31, (Princeton Uni- versit y Press, 1984) [45] R. R. Ko cherlak ota, Integral homology of real ﬂag manifolds and lo op spaces of symmetric spaces, Adv. Math. 110: 1–46 (1995) [46] Y. Ko dama, Y oung diagrams and N -soliton solutions of the KP equation, J. Phys. A 37:11169–11190 (2004) [47] Y. Kodama and K. T-R McLaughlin, Explicit in tegration of the full symmetri c T o da hierarc hy and the sorting property , Lett. Math. Phys. 37:37–47 (1996) [48] Y. Ko dama and V. U. Pierce, Geometry of the Pfaﬀ lattices, Inter. Math. Res. Notes, (2007) rnm 120, 55 pages [49] Y. Ko dama and V. U. Pierce, The Pfaﬀ lattice and the symplectic eigenv alue problem, (2008) [50] Y. Ko dama and J. Y e, T o da hierarchy with indeﬁnite metric, Physica D 91:321–339 (1996) [51] Y. Ko dama and J. Y e, T o da lattices with indeﬁnite metric I I: top ology of the iso-sp ectral manifolds, Ph ysica D 121:89–108 (1998) [52] Y. Kodama and J. Y e, Iso-sp ectral deformations of general matrix and their reductions on Lie algebras, Commun. Math. Phys. 178:765-788 (1996) [53] B. M. Kostant, On Whittaker vectors and representation theory , Inv ent. Math. 48:101–184 (1978) [54] B. M. Kostant, The solution to a generalized T o da lattice and representation theory , Adv. Math. 34:195–338 (1979) [55] P . D. Lax, Integrals of nonlinear equations of evolution and solitary wa ves, Comm. Pure Appl. Math. 21:467–490 (1968) [56] S. V. Manako v, Complete integrabilit y and sto chastization of discrete dynamical systems, So v. Phys. ZETP 40: 269–274 (1975) [57] T. Miwa, M. Jimbo and E. Date, Solitons: Diﬀer ential e quations, symmetries and inﬁnite dimensional algebr as , Cambridge T racts in Mathematics 135 (Cambridge Universit y Press, 2000) [58] J. Moser, Finitely many mass points on the line under the inﬂuence of an exp onential p otential – an integrable system, In: Dynamical Systems, Theory and Applications, Lecture Notes in Ph ysics, V ol. 38, Springer, 1975, p. 467–497, 1975 [59] T. Oda, Con vex Bo dies and Algebraic Geometry: an In tro duction to the Theory of T oric V arieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, V ol. 15, Springer-V erlag (1988) [60] A. M. Perelomov, Inte gr able systems of classic al me chanics and Lie algebr as , (Birkh¨ auser, Basel-Boston-Berlin, 1990) [61] A. G. Reyman, Integrable Hamiltonian systems connected with graded Lie algebras, Zap. Nauch. Sem. LOMI 95:3–54 (1980) [62] A. G. Reyman and M. A. Semenov-Tian-Shan tsky , Reduction of Hamiltonian systems, aﬃne Lie algebras and Lax equations, Inv ent. Math. 54:81-100 (1979) [63] A. G. Reyman and M. A. Semenov-Tian-Shansky , Group-theoretical metho ds in the theory of ﬁnite-dimensional integrable systems, Encyclop ae dia of Mathematical Scienc es , V ol. 16, “Dynamical Systems VII” (Springer-V erlag, Berlin Heidelberg, 1994) 50 YUJI KOD AMA 1 AND BARBARA SHIPMAN 2 [64] M. Sato, Soliton equations as dynamical systems on an inﬁnite dimensional Grassmann manifolds, RIMS Kokyuroku (Kyoto University), 439: 30–46 (1981) [65] B. A. Shipman, On the geometry of certain isosp ectral sets in the full Kostant-T o da lattice, Pac. J. Math. 181(1):159–185 (1997) [66] B. A. Shipman, Mono dromy near the singular level set in the S L (2 , C ) T o da lattice, Phys. Lett. A 239:246–250 (1998) [67] B. A. Shipman, A symmetry of order tw o in the full Kostant-T o da lattice, J. Alg. 215:682–693 (1999) [68] B. A. Shipman, The geometry of the full Kostant-T o da lattice of sl (4 , C ), Journal of Geometry and Physics 33:295-325 (2000) [69] B. A. Shipman, On the Fixed Poin ts of the T o da Hierarch y , Contemp. Math. 285:39–49 (2001) [70] B. A. Shipman, Nongeneric ﬂows in the full T o da lattice, In: Contemporary Mathematics: In tegrable Systems, T op ology , and Physics 309:219–249, American Mathematical So ciety (2002) [71] B. A. Shipman, On the ﬁxed-p oint sets of torus actions on ﬂag manifolds, J. Alg. Appl. 1(3):1–11 (2002) [72] B. A. Shipman, Compactiﬁed isospectral sets of complex tridiagonal Hessen b erg matrices, In: Dynamical Systems and Diﬀerential Equations, Eds. W. F eng, S. Hu, and X. Lu., American Institute of Mathematical Sciences, 788– 797 (2003) [73] B. A. Shipman, A unip otent group action on a ﬂag manifold and ”gap sequences” of p ermutations, Journal of Algebra and Its Applications 2(2):215-222 (2003) [74] B. A. Shipman, Fixed p oints of unip otent group actions in Bruhat cells of a ﬂag manifold, JP Journal of Algebra, Number Theory & Applications 3(2):301-313 (2003) [75] B. A. Shipman, On the Connectedness of a Centralizer, Applied Mathematics Letters 20:467-469 (2007) [76] W. W. Symes, Hamiltonian group actions and integrable systems, Physica 1D:275–280 (1980) [77] W. W. Symes, The QR algorithm and scattering for the ﬁnite nonperio dic T oda lattice, Physica D:275–280 (1982) [78] M. T o da, Vibration of a chain with nonlinear interaction, J. Phys. So c. Japan 22(2):431–436 (1967) [79] M. T o da, W ave propagation in anharmonic lattices, J. Phys. Soc. Japan 23(3):501–506 (1967) [80] C. T omei, The top ology of isospectral manifolds of tridiagonal matrices, Duke Math. J. 51(4):981–996 (1984) Dep ar tment of Ma thema tics, Ohio St a te University, Columbus, OH 43210 E-mail address : kodama@math.ohio-state.edu Dep ar tment of Ma thema tics, The University of Texas a t Arlington, Arlington TX E-mail address : bshipman@uta.edu

The Finite Non-periodic Toda Lattice: A Geometric and Topological Viewpoint

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