The Pfaff lattice on symplectic matrices

The Pfaff lattice on s ymple ctic matrices Y uji Ko dama 1 and Vir gil U. Pierc e 2 1 Department of Mathematics, Ohio State Univ ersity , Colum bus, OH 43210 E-mail: koda ma@mat h.ohio-state.edu 2 Department of Mathematics, Univ ersity of T ex a s – Pan American, E din burg, TX 78539 E-mail: pier cevu@u tpa.edu Abstract. The Pfaff lattice is an integrable system arising from the SR-g roup factorization in an analogous w ay to how the T o da lattice arises from the QR-group factorization. In our recent paper [ Intern. Math. R es. Notic es , (200 7) rnm120], we studied the Pfaff lattice hiera rch y for the case where the La x ma trix is defined to b e a low er Hessenber g matrix. In this paper we deal with the case of a s y mplectic low er Hessenberg La x matrix, this forces the Lax matrix to ta ke a tridiag o nal shap e. W e then s how that the o dd members of the P fa ff lattice hier arch y are trivia l, w hile the even members are equiv alent to the indefinite T o da la ttice hierarchy defined in [Y. Ko dama and J. Y e, Physic a D , 91 (19 96) 321-339]. This is analog o us to the cas e of the T oda lattice hierarch y in the r elation to the Ka c-v an Mo erb eke system. In the case with initial matrix having only real or imag inary eigenv alues, the fixed points of the even flows are given by 2 × 2 blo ck diag o nal matr ic e s with zer o diag onals. W e a ls o consider a family of s kew-orthog onal po lynomials with symplectic recur sion r elation related to the Pfaff lattice, and find that they a re succinctly expr essed in ter ms o f orthogo nal po lynomials app earing in the indefinite T oda la ttice. AMS classification sc heme n um b ers: 37 K10 (37K 20, 3 7K60, 82 B23) Con t en ts 1 In t ro duction 2 2 Bac kground on SR-factorization 5 3 Pfaff lattice hierarc h y 8 3.1 Odd Pfaff flows of symplectic matrices . . . . . . . . . . . . . . . . . . . 9 3.2 Ev en Pfaff flow s o f symplectic matrices . . . . . . . . . . . . . . . . . . . 11 4 Pfaff lattice vs indefinite T o da latt ic e 13 4.1 The indefinite T o da lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 The τ -functions of the Pfaff and T o da lat t ices . . . . . . . . . . . . . . . 15 4.3 Sk ew-orthog onal and ortho gonal functions . . . . . . . . . . . . . . . . . 17 4.4 Asymptotics of the eve n Pfaff flows . . . . . . . . . . . . . . . . . . . . . 20 App endix: SR-algorithm 21 Ac kno wledgmen ts 24 References 24 1. In tro duction The T o da lattice hierarch y of nonlinear differential equations may b e formulated as the con tinuous limit of the QR-algorithm for dia g onalizing a symmetric matrix [19, 8, 20]. There exists a num b er of generalizations of the QR-algorithm for diag o nalizing matrices while preserving different symmetries, the HR and SR -algorithms are t w o that we will consider in this pa p er [4 , 7, 10, 19]. In pa r ticular the SR-algorithm relies o n SR- factorizations whose go a l is to factor an in vertible matrix into a symplectic matrix and an upp er triangula r matr ix [6]. The Pfaff lattice hierarch y w as recen tly in tro duced to describe the par t it io n functions of the orthogonal and symplectic ensem bles of random matrices, a nd equiv alen tly , to g iv e the ev olution of ske w-orthogonal p olynomials (see [1, 3]). The Pfaff la ttice hierarc h y ma y be view ed as a contin uous limit of the SR- algorithm for diagonalizing a lo w er Hessen b erg matrix, and in fact w as orig inally disco ve red in this con text [22]. W e sho we d in our recen t work [16] t ha t the Pfaff la t tice is an in tegra ble system in the Arnold-Liouville sense, and in the case that the initial conditions are not lo wer Hessen b erg matrices from the symplectic algebra, sp ( n ), w e sho wed that the fixed p oin ts of the Pfaff flow s with an initial condition ha ving real distinct eigen v alues are giv en by 2 × 2 block diagonal matrices with zero diagonals. In this pap er we will consider t he Pfaff lattice flow with an initial condition giv en b y a matrix fro m the symplectic algebra sp ( n ). The Lie algebra sp ( n ) is the set o f 2 n × 2 n matrices satisfying J L + L T J = 0 with J = diag 2 0 1 − 1 0 ! , . . . , 0 1 − 1 0 ! ! = I n ⊗ 0 1 − 1 0 ! . Here diag 2 ( A ) denotes the 2 × 2 block diagonal matrix of A , i.e. diag 2 ( A ) = diag 2 ( A 1 , . . . , A 2 ) with the 2 × 2 blo cks A j in the diago na l of A , and I n is the n × n iden tity matrix. Remark 1.1 Matric es in the symple ctic a lgebr a, sp ( n ) , ar e c a l le d Ham i l toni a n in the line ar algebr a and c ontr ol the ory liter atur e, we w il l avo id this nom enclatur e to avoid c onfusion with the Hamiltonia ns which ar e c onstants of motion and app e ar thr oughout the p a p er. The term symple ctic algebr a is also the o ne c ommo n ly use d in the liter atur e on the Pfaff lattic e and in r epr esentation the ory [11]. The sp ectrum of a real symplectic matrix comes in three t yp es: (a) Pairs of real eigen v alues ( z , − z ). (b) Paris of imaginary eigen v alues ( z , − z ). (c) Or quadruples o f complex eigen v alues ( z , ¯ z , − z , − ¯ z ). Symplectic matrices app ear in a n um b er o f con trol theory pro blems most notably in solving the real algebraic Riccati equation (see e.g. [9]). Our J is a p erm utation of the traditional J , sa y ˜ J , used in the literature o n the Hamiltonian eigen v alue problem, i.e. ˜ J = 0 1 − 1 0 ! ⊗ I n . W e use this J b ecause of its connection to the Pfa ff lattice whic h is defined on low er Hessen b erg mat r ices (see b elow ). The Pfaff lattice can b e view ed as an sp -v ersion of the T o da la ttice, it has the follo wing form with sp -pro jection [1, 16], ∂ L ∂ t = [ π s p ( ∇ H ) , L ] . The Pfaff lattice is asso ciated with the Lie-P oisson structure induced by the splitting sl (2 n ) = k ⊕ sp ( n ). Recall that the splitting giv es t he follow ing pro j ections o f a n elemen t X ∈ sl (2 n, R ), π k ( X ) = X − − J ( X + ) T J + 1 2 ( X 0 − J X T 0 J ) (1.1) π s p ( X ) = X + + J ( X + ) T J + 1 2 ( X 0 + J X T 0 J ) (1.2) where X 0 is the 2 × 2 blo c k diagonal part of X , X + (resp. X − ) is the 2 × 2 blo ck upp er (resp. lo wer) triangular part of X . In particular, w e ha v e k =               h 1 I 2 0 2 . . . 0 2 ⋆ h 2 I 2 . . . 0 2 . . . . . . . . . . . . ⋆ ⋆ · · · h n I 2      : n X k =1 h k = 0          , (1.3) where I 2 is the 2 × 2 iden tit y matrix, and 0 2 is the 2 × 2 zero matrix. As in the case of t he T o da la t tice hierarc h y , one can a lso define the Pfaff lattice hierarc hy (see [16], a nd also Section 3), ∂ L ∂ t j = [ π s p ( ∇ H j ) , L ] for j = 1 , . . . , 2 n − 1 , (1.4) where H j = 1 j + 1 tr( L j + 1 ) a nd L is a Hessen b erg matrix g iven b y , for example with n = 3 , L =          0 c 1 0 0 0 0 ∗ b 1 a 1 0 0 0 ∗ ∗ − b 1 c 2 0 0 ∗ ∗ ∗ b 2 a 2 0 ∗ ∗ ∗ ∗ − b 2 c 3 ∗ ∗ ∗ ∗ ∗ 0          . (1.5) The c k are in v arian ts of the Pfaff lattice, in the t r a ditional Pfaff lattice consider b y Adler et. al. [1] the c k are c hosen to b e 1. It w as show n in [16] that the c k are Casimirs of the Pfa ff lat t ice. In [16] it was sho wn that for L with distinct eigen v alues the Pfa ff lattice hierarc hy is an in tegrable system in the Arnold-Liouville sense. In addition it w a s sh o wn that if L is not symplectic and has distinct real eigen v alues, then as t j → ∞ for any j , there is a diagonal matrix P ( t j ) suc h that the no r malized matrix ˆ L ( t j ) := P ( t j ) L ( t j ) P ( t j ) − 1 approac hes a 2 × 2 blo ck upp er triangular matr ix, where the diagonal blo cks are sorted b y the siz e of z j for the real eigen v alues z app earing in eac h. In this pa p er, we deal with the Pfaff latt ice when the initial matrix is symplectic, and discuss the dynamical structure in a connection with the T o da lattice. A natural question at this p oint is then what happ ens when w e restrict the Pfaff lattice v ariables to matrices whic h are b oth sym plectic and low er Hesse n b erg. The first consequenc e is that suc h matr ices ha v e the compact form: L 0 =             0 c 1 d 1 0 0 0 a 1 0 · · · 0 2 0 0 a 1 0 0 c 2 d 2 0 · · · 0 2 . . . . . . . . . . . . 0 2 0 2 · · · 0 c n d n 0             , (1.6) that is, L 0 has a 2 × 2 blo c k tridiagonal form L 0 = ( l i,j ) 1 ≤ i,j ≤ n with 2 × 2 blo c k matrices l i,j ha ving l i,j = 0 2 for | i − j | > 1 and l k ,k = 0 c k d k 0 ! , l k ,k +1 = l k +1 ,k = 0 0 a k 0 ! , where c k = ± 1. W e call a matrix in this form H S -tridiagonal. This is analogous to restricting the T o da lattice v ariables to a matrix whic h is tridiago nal and sk ew- symmetric (i.e. in so ( n )); a situation w e will remark on in more detail in Section 4. W e will show that the ev en Pfaffia n flow of an H S -tridiagonal matr ix con v erges to a sequence o f blo c k diagonal mat r ices with the follo wing shap es: (a) a 2 × 2 blo ck for eac h real pa ir of eigen v alues ( z , − z ), (b) a 2 × 2 blo c k for each imaginary pair of eigen v alues ( z , − z ), (c) a 4 × 4 blo c k for each quadruple of complex eigen v alues ( z , ¯ z , − z , − ¯ z ). In additio n, for the t 2 j flo w, the blo ck s ar e sorted b y t he size of Re( z 2 j ). In t he app endix, w e will give a simple pro of that the SR- algorithm is giv en by the integer ev aluation of the flow generated b y the Pfaff latt ice introduced in [1] (see a lso [16]). The presen t pa p er is or g anized as follow s: In Section 2 we review the SR-f a ctorization whic h is closely asso ciated with the Pfaff lattice hierarch y o f equations. W e then sho w, in Section 3, that the o dd mem b ers o f the hierarc h y of the Pfaff lattice (1 .4) on symplectic matrices are trivial, while the ev en mem b ers of the hierarch y on H S -tridiagonal matrices are equiv alen t to the indefinite T o da hierarch y in tro duced in [17] (Theorem 3.2). This is similar to the case o f the T o da lattice in the conne ction with the Kac-v an Mo erb ek e system. In contrast with the symmetric T o da lattice hierarch y , the indefinite T o da hierarc h y exp eriences blow ups, where some en tries of the ma t r ix approac h infinit y . Our result then implies that fo r generic H S -tridiagonal matr ices t he Pfaff lat tice has blow ups. In Section 4, w e g ive a f urther discussion on the equiv alence b et w een the Pfaff lattice on H S -tridiagonal matrix and the indefinite T o da la ttice defined in [17]. W e b egin by reviewing the indefinite T o da lattice, then sho w that the τ -functions for the Pfaff lattice a r e the same as those for the indefinite T o da lat t ice (Theorem 4.1). W e then examine the families of sk ew-orthogonal p olynomials a pp earing in t he Pfaff lattice for an H S -tridiagonal matrix. W e sho w t ha t these p olynomials a re related to the orthogonal p olynomials in the indefinite T o da lattice (Theorem 4 .2). Finally , w e remark on the asymptotic b ehavior of the ev en Pfaff lattice flo ws o f an H S -tridiagonal ma t r ix. In particular we sho w that initial conditions with complex eigen v alues, or with complex eigen ve ctors, will result in a blo w up. 2. Bac kground on SR-factorization The Pfaff lattice equation is intimately connected to the SR-factor ization of an in v ertible matrix and so w e collect here some relev an t facts on this factorization. With t he Lie subalgebra k giv en b y ( 1 .3) in the Lie algebra splitting sl (2 n ) = k ⊕ sp ( n ), w e define G k to b e the Lie group with Lie algebra k , G k :=               α 1 I 2 0 2 · · · 0 2 ⋆ α 2 I 2 · · · 0 2 . . . . . . . . . . . . ⋆ ⋆ · · · α n I 2      : n Y j = 1 α j = 1          . W e also define the group ˜ G ⊃ G k to b e the group of real in v ertible low er 2 × 2 blo c k triangular matrices with fr e e in v ertible diagonal blo c ks, t ha t is, ˜ G is a para b olic subgroup of lo w er 2 × 2 blo ck matrices of S L (2 n, R ). W e define the Pfaffian o f a sk ew-symmetric matrix m by the recursiv e for mula pf( m ) = 2 n X j = 1 ( − 1) i + j +1 m ij pf( m ˆ i ˆ j ) , where m ˆ i ˆ j is found by deleting the i -th and j - th ro ws and columns of m , where i is any ro w, and the initial v alue for the recursion is give n by pf 0 m 12 − m 12 0 ! = m 12 . Then, for example, with the ske w symmetric matrix m =      0 m 12 m 13 m 14 0 m 23 m 24 0 m 34 0      w e hav e pf( m ) = m 12 m 34 − m 13 m 24 + m 14 m 23 . Throughout this pap er, w e leav e blank t he lo w er triangular part of sk ew-symmetric matrices. W e will use the SR-factorization of g ∈ S L (2 n, R ) introduced by Bunse-Gerstner (Theorem 3.8 in [6], see a lso [5]): Theorem 2.1 L et g ∈ S L (2 n, R ) and M = g J g T . Then the factorization g = r s with r ∈ ˜ G a n d s ∈ S p ( n, R ) exists if and only if the Pfaffian of the 2 k × 2 k upp e r left submatrix of M , denote d by M 2 k , do es not vanish, i.e. pf ( M 2 k ) 6 = 0 , 1 ≤ k ≤ n − 1 . As a consequence there is a dens e set of matrices in S L (2 n, R ) for whic h this decomp osition exists. If w e restrict the factorization to using elemen ts o f G k ⊂ ˜ G the Theorem b ecomes: Theorem 2.2 L et g ∈ S L (2 n, R ) and M = g J g T . Then the factorization g = r s with r ∈ G k and s ∈ S p ( n, R ) ex ists if and only if al l the Pfa ffians pf ( M 2 k ) satisfy the p ositivity c ondition, pf ( M 2 k ) > 0 , 1 ≤ k ≤ n − 1 . This implies that there is a n o p en set of matrices in S L (2 n, R ) for whic h this decomp osition exists. Theorem 3.7 in [6] sho ws that the set o f matrices in S L (2 n, C ) with SR-factorization is neither op en nor dense in S L (2 n, C ). F or this reason w e will restrict our consideration to the real groups for now. In Remark 3.9 in [6] it is noted that for the c hoice of subgroups of S L (2 n, R ) in that pap er t he decomp osition g = r s is not unique, as o ne ma y pass a factor from ˜ G ∩ S p ( n, R ) = { diag 2 ( A 1 , A 2 , . . . , A n ) : det( A j ) = 1 ∀ j } , b et w een r and s . In the case of S R factorizat io n with R restricted to G k w e find that factorizations are unique up to a factor fr o m G k ∩ S p ( n, R ) = { diag 2 ( ± I 2 , ± I 2 , . . . , ± I 2 ) } . One notes t hat G k is in fact made up of 2 n connected comp onents , and if w e further restrict R to the comp onen t of G k con ta ining the identit y matrix w e obtain unique factorizations. Equiv alen tly w e obta in unique factorizations b y a sking that the diag onal elemen ts of r b e p o sitiv e. In con trast the group ˜ G is connected. The Pfaffians defined in Theorem 2.1 a re a lso called the τ -functions, whic h play the fundamen tal role in the Pfaff latt ice hierarc hy [1, 16], i.e. for M = g J g T and M 2 k , the 2 k × 2 k upp er left submatrix of M , τ 2 k := pf ( M 2 k ) , k = 0 , 1 , . . . , n , (2.1) with τ 0 = 1. Deriv atives of the τ -functions f or g ( t ) := exp  P 2 n − 1 j = 1 t j L j 0  , with L (0) = L 0 , generate the matrix en tries of L ( t ) in the form (1.5), and the solution of the Pfaff latt ice hierarc h y . F or example, w e ha v e [1, 16] (see Section 3), a k ( t ) = a k (0) p τ 2 k + 2 ( t ) τ 2 k − 2 ( t ) τ 2 k ( t ) , b k ( t ) = ∂ ∂ t 1 ln τ 2 k ( t ) . (2.2) W e also ha v e the fo llo wing Lemma: Lemma 2.1 L et g = r s b e a 2 n × 2 n ma trix wh e r e r ∈ G k and s ∈ S p ( n, R ) . Then the 2 × 2 blo ck dia g onal entries of r , diag 2 ( r ) = diag 2 ( r 1 I 2 , . . . , r n I 2 ) , ar e expr esse d by the τ -functions, r k = r τ 2 k τ 2 k − 2 . Pr o of . Since M = g J g T = r sJ s T r T = r J r T , w e ha ve τ 2 k = pf ( M 2 k ) = Q k j = 1 r 2 j , whic h leads to the formula. A necessary condition for the existence of the factorization is that τ 2 k = pf ( M 2 k ) m ust b e p o sitive (Theorem 2.2); Lemma 2.1 and t he con v en tion that τ 0 = 1 explain this condition. There are explicit algorithms f o r carrying out SR-f actorization a nd they ha v e man y features in common with those dev elop ed for QR-factorization, w e refer the reader to [9] and [6]. 3. Pfaff lattice hierarc hy In [1], Adler, Horozo v, and v an Mo erb ek e introduced the Pfaff lattice hierarc h y to describe the partition functions of the Gaussian o r thogonal and symplectic ensem bles of random matrices (GOE and GSE), and to describ e the ev olution of the recursion relations of sk ew-orthog o nal p olynomials. The finite Pfaff lattice hierarc h y is a set of Hamiltonian flows on sl (2 n, R ) with the Lie- P oisson structure induced b y the splitting sl (2 n, R ) = k ⊕ sp ( n ), and it is defined a s follow s ( see also [16]): F or an y X , Y ∈ sl (2 n ) w e first define the R -brac k et [ X , Y ] R = [ R X, Y ] + [ X , R Y ] , where R is the R - ma t rix giv en b y R = 1 2 ( π k − π s p ). Then the Lie-P oisson brack et f o r an y functions F and H on sl ∗ (2 n ) ∼ = sl (2 n ) is defined b y { F , H } R ( L ) = h L, [ ∇ F , ∇ H ] R i , where h A, B i = tr( AB ) and ∇ F is defined by h X, ∇ F i = d dǫ F ( L + ǫX ) | ǫ =0 . The Pfaff lattice with resp ect to a Hamiltonian function H ( L ) is defined by dL dt = { H ( L ) , L } R ( L ) . In particular, if the Hamiltonian is S p -in v arian t, one can write { H ( L ) , L } R ( L ) = [ π s p ( ∇ H ( L )) , L ] . The traditional Hamiltonians giving the Pfaff lattice are 1 k +1 tr( L k +1 ) [16]. The Pfaff lattice hierarc h y is then defined b y ∂ L ∂ t k = [ π s p ( L k ) , L ] k = 1 , . . . , 2 n − 1 . (3.1) Th us the t k -flo w of the Pfaff lattice hierarc hy is asso ciated to t he Hamiltonian H k = 1 k +1 tr( L k +1 ). Each flo w is solve d b y the follo wing factorization pro cedure: F actor exp ( t ∇ H ( L (0))) = R ( t ) S ( t ) (3.2) with the initial matrix L (0) using SR-factorization with R in the connected comp onen t of G k con ta ining the identit y a nd S ∈ S p ( n ), then the solution is giv en by L ( t ) = R ( t ) − 1 L (0) R ( t ) = S ( t ) L (0) S ( t ) − 1 . (3.3) With this in mind, one sees t ha t as in the case of the QR-algor it hm with the T o da lattice, the SR-alg orithm is giv en b y in teger ev alua t io ns of the Pfaff flow with Hamiltonian H ( L ) = tr( L ln( L ) − L ) a result w e will sho w in detail in the App endix. In [16], w e show ed that the Pfaff latt ice hierarc hy (3.1) is an in tegrable system in the Arnold-Liouville sense. Normalizing the matrix L o f the form (1.5) b y ˆ L = P LP − 1 with the diago na l matrix P in the 2 × 2 blo ck form, P = diag 2 I 2 , a 1 I 2 , ( a 1 a 2 ) I 2 , . . . , n − 1 Y j = 1 a j ! I 2 ! , ˆ L is a low er Hessen b erg matrix with 1’s on the sup er diagonal. Then we show ed that if L w as not symplectic and a ll the eigen v alues are r e al and distinct , then as t j → ∞ , ˆ L ( t j ) conv erges to a 2 × 2 blo c k upp er triangular ma t rix suc h that the diagona l blo cks are sorted by the size of z j . Solutions of t he Pfaff lattice hierarc hy a re generated b y the τ -functions (in tro duced as obstructions to the SR-factorizat io n in the prev ious section). They are found by the follo wing pro cedure: W e first consider the factorization o f g ( t ) := exp  P 2 n − 1 j = 1 t j L j 0  with L 0 = L (0) (see (3.2)) , g ( t ) = R ( t ) S ( t ) with R ( t ) ∈ G k , S ( t ) ∈ S p ( n ) . Then the ske w-symmetric matrix M ( t ) = g ( t ) J g T ( t ) b ecomes M ( t ) = R ( t ) J R T ( t ) . Since R ∈ G k , w e hav e diag 2 ( R ) := diag ( r 1 I 2 , r 2 I 2 , . . . , r n I 2 ) . Then the τ - functions τ 2 k defined in ( 2 .1) can b e written b y τ 2 k ( t ) = pf ( M ( t ) 2 k ) = k Y j = 1 r j ( t ) 2 . Then from (3.3 ), i.e. R ( t ) L ( t ) = L (0) R ( t ), w e hav e a k ( t ) = a k (0) r k +1 ( t ) r k ( t ) = a k (0) p τ 2 k + 2 ( t ) τ 2 k − 2 ( t ) τ 2 k ( t ) , whic h giv es the a k ’s in (2.2) (see also [1, 16]). 3.1. Odd Pfaff flows of sympl e ctic m atric es Here w e sho w that if L (0) is a symplectic matrix then it is a fixed p oin t o f the o dd mem b ers of the Pfaff lattice hierarc hy . T o see this, one no t es: Lemma 3.1 F or L ∈ sp ( n ) , the o d d p ower L 2 j − 1 is also symple ctic. Pr o of . Being symplectic is equiv alent to J LJ = L T . Supp ose that J L 2 j − 3 J =  L T  2 j − 3 . Then w e ha v e J L 2 j − 1 J = J L 2 j − 3 J J LJ J LJ =  L T  2 j − 3 L T L T =  L T  2 j − 1 so that the lemma is true b y induction. Therefore, π s p ( L 2 j − 1 ) = L 2 j − 1 for L ∈ sp ( n ), hence the o dd mem b ers of the Pfaff hierarc hy b ecome tr ivial, i.e. ∂ L ∂ t 2 j − 1 = [ L 2 j − 1 , L ] = 0 . Note in pa rticular that all b k in (2.2) v anish, whic h is consisten t with the f o rm L in Theorem App endix A.5, that is, the diagonal elemen ts are all zero. W e note that this also happ ens for the T o da lattice hierar ch y: Recall that the T o da la ttice equation for a symmetric matrix is based on the Lie algebra splitting sl ( n ) = b ⊕ so ( n ), where b is the set of upp er triangular matrices. With the pairing h A, B i := tr( AB ) fo r A, B ∈ sl ( n ), w e ha ve sl ( n ) ∼ = sl ( n ) ∗ = b ∗ ⊕ so ( n ) ∗ where b ∗ ∼ = so ( n ) ⊥ = Sym( n ) and so ( n ) ∗ ∼ = b ⊥ = n . Here Sym( n ) is the set of symmetric matrices, and n is the set o f strictly upp er triangular matrices. Then t he L ie- P oisson brac ket for the functions F , G on Sym( n ) is defined by { F , H } ( L ) = h L, [ ∇ F, ∇ H ] i for L ∈ Sym( n ) . The T o da lattice hierarc h y fo r L ∈ Sym ( n ) is then defined by ∂ L ∂ t j = { H j , L } ( L ) = [ π s o ( ∇ H j ) , L ] . (3.4) With the Hamiltonian functions H j ( L ) = 1 j + 1 tr( L j + 1 ) for j = 1 , . . . , n − 1, the differen t ia l equation for L b ecomes ∂ L ∂ t j = [ π s o ( L j ) , L ] , (3.5) with π s o ( X ) meaning the pro jection of the matrix X on the so ( n ) comp onen t. No w if w e extend the T o da lattice equation for the g eneral L ∈ sl ( n, R ), an analogue of Lemma 3.1 is true with so ( n ) instead of sp ( n ). Then, for L ∈ so ( n ) (i.e. sk ew- symmetric), the o dd mem b ers of the hierarc h y b ecome trivial. How ev er the ev en p o w ers of L are symmetric, and the ev en members of the generalized T o da hierarc h y for L are equiv alen t to symmetric tridiagonal T o da la ttices. This case includes the Kac-v an Mo erb ek e system [14]: Consider for example the follow ing 2 n × 2 n sk ew-symmetric tridiagonal matrix, L =        0 α 1 0 · · · 0 − α 1 0 α 2 · · · 0 . . . . . . . . . . . . . . . 0 · · · − α 2 n − 2 0 α 2 n − 1 0 · · · 0 − α 2 n − 1 0        . Then the Kac-v an Mo erb ek e hierarc h y may b e expressed b y ∂ L ∂ t 2 j = [ π s o ( L 2 j ) , L ], where the first mem b er giv es ∂ 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 . W e then not e that the square L 2 is a symmetric matrix, L 2 = T (1) ⊗ 1 0 0 0 ! + T (2) ⊗ 0 0 0 1 ! , (3.6) where T ( i ) , for i = 1 , 2, are n × n symmetric tridiagona l ma t rices giv en by 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 . Then one can sho w tha t eac h T ( i ) giv es the symme tric T o da lattice, t ha t is, the equation ∂ L ∂ t 2 j = [ π s o ( L 2 j ) , L ] splits in to tw o T o da lattices, ∂ T ( i ) ∂ t 2 j = [ π s o ( T ( i ) ) j , T ( i ) ] for i = 1 , 2 . The equations for T ( i ) are connected by the Miura-type transfor ma t io n, with the functions ( a ( i ) k , b ( i ) k ), through the Kac-v an Mo erb ek e v ariables α k (see [13]). W e now sho w that the ev en Pfaff flows on a H S -tridaigonal matrix ha v e a similar structure. 3.2. Even Pfaff flows of symple c tic matric es W e first show that the H S -tridiagonal form (1 .6) is in v arian t under the ev en mem b ers of the Pfaff hierarc h y . Then we sho w that the Pfaff flow for the matrix L in the form (1.6) is related to the indefinite T o da la ttice defined in [1 7 ]. The indefinite T o da lat t ice is a con tinuous ve rsion of the HR algo rithm [10], and is defined as follows (see also Section 4 for some details): Let C b e the n × n diagonal matrix giv en by , C = diag( c 1 , c 2 , . . . , c n ) , where c k ’s a r e the elemen ts app earing in the diagonal blo cks of (1 .6), and they ar e in v aria nt under the Pfaff flo w (see [16]). W e also define ˜ T b y ˜ T := C T with the triadiagonal matrix (A.2), i.e. ˜ T =        c 1 d 1 c 1 a 1 · · · · · · 0 c 2 a 1 c 2 d 2 · · · · · · 0 . . . . . . . . . . . . . . . 0 0 · · · c n − 1 d n − 1 c n − 1 a n − 1 0 0 · · · c n a n − 1 c n d n        . (3.7) Then the indefinite T o da la ttice hierarc h y is defined by ∂ ˜ T ∂ t 2 j = [ B j , ˜ T ] , (3.8) where B j := [ ˜ T j ] + − [ ˜ T j ] − . Here the [ ˜ T j ] ± represen t the pro jections on the upp er (+) and low er ( − ) tria ngular parts of t he matrix ˜ T j (sometimes we write ˜ T j ± = [ ˜ T j ] ± ). With L in the H S -tridiagonal fo rm (1.6), w e first note L 2 = ˜ T ⊗ 1 0 0 0 ! + ˜ T T ⊗ 0 0 0 1 ! , (3.9) from whic h we ha ve, for any ev en p ow er, L 2 j = ˜ T j ⊗ 1 0 0 0 ! + ( ˜ T j ) T ⊗ 0 0 0 1 ! . Remark 3.1 We r emark her e that e quation (3.9) is similar to (3.6), this structur e is fundamental and implies that our pr oblem wil l have a sim ilar structur e to that of the Kac-van Mo erb eke lattic e. Then using the pro jection (1 .2), w e obtain π s p ( L 2 j ) = B j ⊗ 1 0 0 0 ! − B T j ⊗ 0 0 0 1 ! , (3.10) with B j = ˜ T j + − ˜ T j − . With ( 3 .10), one can easily sho w tha t the H S -tridiagonal form is in v aria nt under the ev en mem b ers of the Pfaff lattice hierarc hy . It is also imme diate to see from (3 .10) that the 2 m -th mem b er of the Pfaff lattice hierarc hy (1.4 ) in terms of L 2 is equiv alen t to the indefinite T o da lattice (3.8). Th us we ha v e the following theorem: Theorem 3.2 The e v en Pfaff flow for L ∈ sp ( n, R ) in the H S -tridiagonal matrix form (1.6), ∂ L ∂ t 2 m =  π s p  L 2 m  , L  , is e quivalent to the ind e finite T o da flow for ˜ T = C T with the symmetric tridiagonal matrix T o f (A.2) and C := diag( c 1 , c 2 , . . . , c n ) , ∂ ˜ T ∂ t 2 m = [ B m , ˜ T ] , (3.11) wher e B m = ˜ T m + − ˜ T m − . If the c k = 1 for all k , then ˜ T = T is symmetric and B m = π s o ( T m ). Theorem 3.2 then say s that the 2 m -th Pfaff flo w is just the traditional symmetric tridiagonal m -th flo w of the T o da lattice hierarch y . In terms of the SR-algorithm in Prop osition App endix A.4, w e obtain as a Corollary of Theorem 3.2 that the even -inte ger iterates, L 2 k of the SR-algorithm are equiv alen t to the ˜ T k iterates o f the HR- algorithm. This result is sho wn directly in [4], the k ey is t he structure o f equation (3.9). With the p erm uted J matrix used in [4], i.e. J = 0 1 − 1 0 ! ⊗ I n , the righ t hand side of relation (3.9) take s the form ˜ T 0 n 0 n ˜ T T ! = 1 0 0 0 ! ⊗ ˜ T + 0 0 0 1 ! ⊗ ˜ T T . 4. Pfaff lattice vs indefinite T o da lattice Here w e giv e a further discuss ion on the equiv alence betw een the Pfaff lattice hierarc h y on H S -tridiagonal matrix and the indefinite T o da lattice hierarch y intro duced in [1 7]. 4.1. The indefi nite T o da lattic e Let us b egin with a brief description of the indefinite T o da latt ice defined b y (3.8) for a matrix ˜ T = C T with a symmetric matrix T of the form (A.2), i.e. ∂ ˜ T ∂ t = [ B , ˜ T ] with B = ˜ T + − ˜ T − . The solution ˜ T ( t ) with the initial matrix ˜ T 0 can b e solved b y the HR-f a ctorization, ˜ g = exp( t ˜ T 0 ) = r ( t ) h ( t ) , where r is a low er triangular matrix and h satisfies hC h T = C (recall t ha t if C = I n , h ∈ S O ( n )). It is then easy to show that the solution ˜ T ( t ) is obtained by ˜ T ( t ) = r ( t ) − 1 ˜ T 0 r ( t ) = h ( t ) ˜ T 0 h ( t ) − 1 . (4.1) W e not e that the en tries a k in ˜ T ar e expressed in terms of the diago nal elemen ts o f r , sa y diag ( r ) = diag( r 1 , . . . , r n ), a k ( t ) = a k (0) r k +1 ( t ) r k ( t ) . T o find those diagonal en tries from ˜ g , o ne considers the follo wing matrix called the momen t matrix, M T oda := ˜ g C ˜ g T = r C r T . (4.2) Then the matrix r can b e found by the Cholesky factorization metho d. Now the τ - functions are defined b y τ T oda k = det ( M T oda k ) , where M T oda k is t he k × k upp er left submatrix of M T oda . With (4.2), w e hav e τ T oda k = Q k j = 1 c j r 2 j , which giv es c k r 2 k = τ k /τ k − 1 . Then the en tries a k of ˜ T can b e expresse d in terms of the τ -functions, a k ( t ) = a k (0) r c k c k +1  τ T oda k +1 ( t ) τ T oda k − 1 ( t )  1 / 2 τ T oda k ( t ) . (4.3) As will b e sho wn in the next section (see Theorem 4.1 b elow), those a r e the same a s the formulae for the a k ’s in (2.2) giv en in terms of the τ - functions of t he Pfaff lattice. Let us also discuss the orthogonal functions app earing in the indefinite T o da latt ice. First w e note that the Lax form (3.8) is give n by the compatibilit y of the equations, ˜ T Φ = Φ D and ∂ Φ ∂ t 2 j = B j Φ , where D = diag( λ 1 , . . . , λ n ) with the eigen v alues λ k , and Φ = ( φ i ( λ j )) 1 ≤ i,j ≤ n the eigenmatrix. As the orthogona lity condition of the eigen v ectors with a norma lizat io n, w e hav e Lemma 4.1 Φ C Φ T = C . Pr o of . W e consider the follow ing relation whic h is equiv alen t to the orthogona lit y relation, Φ T C − 1 Φ = C − 1 . (This is sometimes called the second orthog onalit y relation.) T o sho w this, w e note λ j φ ( λ i ) T C − 1 φ ( λ j ) = φ ( λ i ) T C − 1 ˜ T φ ( λ j ) = φ ( λ i ) T T φ ( λ j ) = ( ˜ T φ ( λ i )) T C − 1 φ ( λ j ) = λ i φ ( λ i ) T C − 1 φ ( λ j ) . Since we assume λ i 6 = λ j , the matrix Φ T C − 1 Φ is diagonal. W e also note that this matrix is in v aria n t under the flow, i.e. ∂ (Φ T C − 1 Φ) /∂ t = 0. No r ma lizing the diagonal mat rix giv es the result. W e then define the inner pr o duct for f unctions f ( λ ) a nd g ( λ ), h f g i T oda := n X k =1 f ( λ k ) g ( λ k ) c k , (4.4) whic h defines a discrete measure dµ ( λ ) = P n k =1 c k δ ( λ − λ k ) dz . Lemma 4.1 implies h φ i φ j i T oda = c i δ ij . With the ort hogonalit y relation, the momen t matrix M T oda can b e expressed b y M T oda = Φ 0 e 2 tD C Φ T 0 =  h φ 0 i φ 0 j e 2 tz i T oda  1 ≤ i,j ≤ n , (4.5) where Φ 0 is the initial eigenmatrix with the eigen v ector φ 0 i = φ 0 i ( λ ) = φ i ( λ, 0). In terms of the initial eigenmatrix Φ 0 , w e can find an explicit form of Φ( t ) for the indefinite T o da lat t ice hierarc hy with t = ( t 2 , t 4 , . . . , t 2 n ) (see Theorem 1 in [17] and Theorem 2 in [15]): Lemma 4.2 The eigenve ctor φ ( λ , t ) = ( φ 1 ( λ, t ) , . . . , φ n ( λ, t )) T c an b e expr esse d as φ k ( λ, t ) = c 1 / 2 k e ξ ( λ, t )  τ T oda k ( t ) τ T oda k − 1 ( t )  1 / 2          m 1 , 1 ( t ) · · · m 1 ,k − 1 ( t ) φ 0 1 ( λ ) m 2 , 1 ( t ) · · · m 2 ,k − 1 ( t ) φ 0 2 ( λ ) . . . . . . . . . . . . m k , 1 ( t ) · · · m k ,k − 1 ( t ) φ 0 k − 1 ( λ )          , ( 4.6) wher e m i,j ( t ) := h φ 0 i φ 0 j e 2 ξ ( λ, t ) i T oda with ξ ( λ, t ) = P n k =1 λ k t 2 k , and the τ -functions ar e given by τ T oda k = det  M T oda k  = | ( m i,j ) 1 ≤ i,j ≤ k | . (4.7) Pr o of . With the factorization e ξ ( ˜ T 0 , t ) = r ( t ) h ( t ), w e ha ve ˜ T ( t ) = h ( t ) ˜ T 0 h ( t ) − 1 (see (4.1)). Then from Lemma 4.1, w e o btain Φ( t ) = h ( t )Φ 0 . No w from the fa ctorization with ˜ T Φ = Φ D , we ha ve Φ( t ) = r ( t ) − 1 e ξ ( ˜ T 0 , t ) Φ 0 = r ( t ) − 1 Φ 0 e ξ ( D , t ) , whic h implies φ i ( λ, t ) ∈ Span  φ 0 1 ( λ ) e ξ ( λ, t ) , . . . , φ 0 i ( λ ) e ξ ( λ, t )  , i = 1 , . . . , n . (4.8) Then using the Gram-Sch midt orthogonalizatio n metho d with the inner pro duct (4.4), w e obtain the result. If w e set φ 0 1 ( λ ) = 1 and consider the semi-infinite lattice with n = ∞ , φ k ( λ, t ) can b e expressed in the following elegan t form in terms of o nly the τ -functions: Prop osition 4.1 The ortho gonal eigenfunctions φ k ( λ ) c an b e expr esse d in terms of τ - functions, φ k ( λ, t ) = c 1 / 2 k e ξ ( λ, t )  τ T oda k ( t ) τ T oda k − 1 ( t )  1 / 2 τ T oda k − 1  t − 1 2 [ λ − 1 ]  λ k − 1 , wher e ξ ( λ, t ) = P ∞ k =1 λ k t 2 k and τ T oda j  t − 1 2 [ λ − 1 ]  = τ T oda j  t 2 − 1 2 λ , . . . , t 2 n − 1 2 nλ n , . . .  . Pr o of . This prop osition app ears in [2] for the case c k = 1. In this more general case the pro of follo ws from form ula (4.6) with ξ ( λ, t ) = P ∞ k =1 λ k t 2 k : First w e note that using (4.8), one can replace φ 0 j ( λ ) in (4.6) b y λ j − 1 with φ 0 1 ( λ ) = 1, whic h giv es m i,j ( t ) = h λ i + j − 2 e 2 ξ ( λ, t ) i T oda . This implies that m i,j ( t ) = ∂ i + j − 2 m 1 , 1 ( t ) /∂ t i + j − 2 1 , hence τ T oda k ( t ) is giv en b y the Hank el determinan t f orm, τ T oda k = | ( h i + j − 1 ) 1 ≤ i,j ≤ k | with h i + j − 1 := m i,j . Since h j ( t ) is a linear com binatio n of the exp onen tial function E i ( t ) = e 2 ξ ( λ i , t ) and E i ( t − 1 2 [ λ − 1 ]) = (1 − λ i λ ) E i ( t ), w e hav e h j  t − 1 2 [ λ − 1 ]  = h j ( t ) − 1 λ h j + 1 ( t ) . Then it is easy to see that τ T oda k  t − 1 2 [ λ − 1 ]  λ k =            h 1 ( t ) h 2 ( t ) · · · h k ( t ) 1 h 2 ( t ) h 3 ( t ) · · · h k +1 ( t ) λ . . . . . . . . . . . . . . . h k ( t ) h k +1 ( t ) · · · h 2 k − 1 ( t ) λ k − 1 h k +1 ( t ) h k +2 ( t ) · · · h 2 k ( t ) λ k            , whic h giv es the determinan t in (4.7) in terms of τ T oda k − 1 . F or a finite n , one can see that τ T oda n ( t − 1 2 [ λ − 1 ]) λ n /τ T oda n ( t ) is prop o rtional to the p olynomial Q n k =1 ( λ − λ k ), and w e ha ve φ n +1 ( λ ) = 0, whic h is just the c haracteristic p olynomial of L . With Propo sition 4.1, w e will show a further ex ample of the close relation b et w een the orthogona l functions in the indefinite T o da lattice a nd the sk ew-orthogonal functions in the Pfaff la ttice for the symplectic matrix o f (1.6). 4.2. The τ -functions of the Pfaff and T o da lattic es W e sho w here that t he τ -functions whic h generate the solutions o f the Pfaff lattice equations are equiv alen t to the τ -functions which generate the solutions of the indefinite T o da lattice hierarc h y . L et us recall that the τ - functions of the Pfaff lattice are defined b y (2.1), i.e. τ 2 k = pf ( M 2 k ) , where M 2 k is the 2 k × 2 k upp er left submatrix of 2 n × 2 n ske w-symmetric matrix M giv en by M := g J g T with g = e ξ ( L 2 0 , t ) , where ξ ( L 2 0 , t ) = P n k =1 t 2 k L 2 k 0 with the initial matr ix L (0) = L 0 . Then our goal is to sho w the f o llo wing Theorem: Theorem 4.1 The τ functions of the even-flows of the Pfaff lattic e hier ar chy with the initial matrix L (0) in the H S -tridiagonal form of (1.6) ar e r elate d to the τ -functions of the indefinite T o da lattic e hier ar chy by τ 2 k = 1 c 1 · · · c k τ T oda k . Pr o of . W e first note from (3.9) that the entries of g = e ξ ( L 2 0 , t ) are expressed b y those of ˜ g = e ξ ( ˜ T 0 , t ) , g = ˜ g ⊗ 1 0 0 0 ! + ˜ g T ⊗ 0 0 0 1 ! . Since ˜ T 0 = C T 0 with a symmetric matr ix T 0 , C − 1 ˜ g is symmetric, i.e. C − 1 ˜ g = ˜ g T C − 1 . Then for H = C − 1 M T oda = C − 1 ˜ g C ˜ g T = ˜ g T ˜ g T and M = g J g T , w e hav e M = H ⊗ 0 1 0 0 ! + H T ⊗ 0 0 − 1 0 ! . (4.9) Let H k b e the k × k upp er left submatrix of H , and M 2 k b e the submatrix defined ab ov e. Then with the p erm utation matrix P = [ e 1 , e 3 , . . . , e 2 k − 1 , e 2 , e 4 , . . . , e 2 k ] , where e k ’s are the standard basis (column) v ectors on R 2 k , one can tr a nsform M 2 k to P T M 2 k P = 0 k H k − H T k 0 k ! = H k 0 k 0 k I k ! P T J P H T k 0 k 0 k I k ! . (4.10) W e recall the Pfaffian identit y f or a sk ew-symmetric matrix B , pf( A T B A ) = det ( A )pf( B ) , whic h with (4.1 0) implies that det( P )pf( M 2 k ) = det( P )det H k 0 k 0 k I k ! pf( J ) . W e finish the pro o f by noting that P is inv ertible and pf( J ) = 1 giving pf( M 2 k ) = det H k 0 k 0 k I k ! = det ( H k ) = 1 c 1 · · · c k det( M T oda k ) . 4.3. Skew-ortho go nal and ortho gonal functions W e sho w here that the eigenv ectors of the matrix L in the form (1.6) define a family of sk ew-orthogonal f unctions, and then discuss an explicit relation b et wee n those sk ew- orthogonal functions and the or thogonal functions app earing in the indefinite T o da lattice, i.e. those in Prop osition 4 .1. As in t he case of the T o da latt ice, the Pfaff lattice hierarc h y (3.1) is giv en by the compatibilit y of t he linear equations, L Ψ = ΨΛ and ∂ Ψ ∂ t 2 j = π s p ( L 2 j ) Ψ , (4.11) where L is symplectic and in the form of (1.6), and Λ is the eigen v alue matrix, Λ := D ⊗ − 1 0 0 1 ! , with D = diag( z 1 , . . . , z n ). (Recall that the eigen v alues of L consist of the pairs ( z k , − z k ) for k = 1 , . . . , n , and w e a ssume here that they are all distinct.) Then w e ha v e: Lemma 4.3 The eigenma trix Ψ satisfies the sk e w-ortho gonal r elation, Ψ T J Ψ = K J , with a diagon a l m atrix with nonzer o c on s tants κ k ’s in the form, K = diag 2 ( κ 1 I 2 , κ 2 I 2 , . . . , κ n I 2 ) . Pr o of . F rom (4.11) with B := π s p ( L 2 j ) and t = t 2 j , w e hav e ∂ ∂ t (Ψ T J Ψ ) = Ψ T B T J Ψ + Ψ T J B Ψ = Ψ T ( B T J + J B )Ψ = 0 , where w e hav e used B ∈ sp ( n ). No w w e note the following equation for the eigenv ectors ψ ( z k ), z j ψ ( z k ) T J ψ ( z j ) = ψ ( z k ) T J Lψ ( z j ) = − ψ ( z k ) T L T J ψ ( z j ) ( ∵ J L + L T J = 0) = − ( Lψ ( z k )) T J ψ ( z j ) = − z k ψ ( z k ) T J ψ ( z j ) . This implies ( z k + z j ) ψ ( z k ) T J ψ ( z j ) = 0 . Noting the order of the eigenv alues in Λ = diag( − z 1 , z 1 , . . . , − z n , z n ), so that Ψ T J Ψ , whic h is ske w-symmetric, is a m ultiple of J of the form K with nonzero constant κ k ’s. F rom Lemma 4 .3, w e hav e Ψ K − 1 J Ψ T = J , (4.12) whic h implies the sk ew-orthogonality relatio n for the functions ψ k ( z ) in the eigen v ector ψ ( z ) = ( ψ 0 ( z ) , ψ 1 ( z ) , . . . , ψ 2 n − 1 ( z )) of L , ( h ψ 2 j , ψ 2 k i = h ψ 2 j +1 , ψ 2 k + 1 i = 0 , h ψ 2 j , ψ 2 k + 1 i = h ψ 2 j − 1 , ψ 2 k i = δ j k , for j ≤ k . (4.13) Here the inner pro duct h f , g i is defined b y h f , g i := n X k =1 [ f ( − z k ) g ( z k ) − f ( z k ) g ( − z k )] κ − 1 k . (4.14) No w w e claim: Theorem 4.2 The skew-ortho gonal eigenfunctions ψ k ( z , t ) satisfying (4.13) c an b e given by      ψ 2 k ( z , t ) = φ k +1 ( λ, t ) ψ 2 k + 1 ( z , t ) = c − 1 k +1 z φ k +1 ( λ, t ) with λ = z 2 , wher e φ k ( λ, t ) ’s a r e the ortho gonal eigenfunc tion s app e aring in the i n definite T o da la ttic e (se e Pr op osition 4.1), and the me asur e of the inne r pr o duct h· , ·i is given by κ k = 2 z k c − 1 k . Before proving the Theorem, w e recall the following Prop osition whic h gives the sk ew-orthogo na l functions app earing in the semi-infinite Pfaff lattice: Prop osition 4.2 The eige nve ctor ψ ( z , t ) = ( ψ 0 ( z , t ) , ψ 1 ( z , t ) , . . . ) T for the matrix L of (1.5) wi th n = ∞ c an b e expr esse d i n terms of τ -func tion s ,              ψ 2 k ( z , t ) = e ξ ( z , t ) [ τ 2 k ( t ) τ 2 k + 2 ( t )] 1 / 2 τ 2 k ( t − [ z − 1 ]) z 2 k ψ 2 k + 1 ( z , t ) = c − 1 k +1 e ξ ( z , t ) [ τ 2 k ( t ) τ 2 k + 2 ( t )] 1 / 2  z + ∂ ∂ t 1  τ 2 k ( t − [ z − 1 ]) z 2 k , wher e ξ ( z , t ) = P ∞ k =1 z k t k and τ 2 k ( t − [ z − 1 ]) = τ 2 k ( t 1 − 1 z , t 2 − 1 2 z , . . . ) . This Prop osition app ear s as Theorem 3.2 in [3] for the case c k = 1. In this mo r e general case the additiona l factor of a c − 1 k +1 is presen t to ensure that the rec ursion relation L has L 2 k + 1 , 2 k = c k +1 . This en try of the Pfaff v ariable is a Casimir of the Pfaff lattice equations and so is fixed. In the definition of the sk ew-ortho gonal f unctions it corresp onds to a c hoice o f the ratio b etw een leading co efficien ts of the p o lynomial parts of ψ 2 k and ψ 2 k + 1 . No w w e prov e Theorem 4.2: Pr o of . F irst w e sho w that ψ k ( z , t )’s in the Theorem are the same as those in Prop osition 4.2, when the matrix L is in an H S -tridiagonal form (1.6). T o show this, w e recall Theorem 4.1, i.e. τ 2 k ( t ) = 1 c 1 · · · c k τ T oda k ( t ) . Since the o dd flo ws are trivial, w e ha v e t = ( t 2 , t 4 , . . . , t 2 n ). Then substituting t his relation in to ψ k ( z , t ) in Prop osition 4.2, it is straightforw ard to show the equations in the Theorem. No w w e sho w the sk ew-orthogonality relation (4.3): W e only che c k the case h ψ 2 j , ψ 2 k + 1 i = δ j k , and the others are trivial with the form of t he inner pro duct. h ψ 2 j , ψ 2 k + 1 i = n X i =1 [ ψ 2 j ( − z i ) ψ 2 k + 1 ( z i ) − ψ 2 j ( z i ) ψ 2 k + 1 ( − z i )] c i 2 z i = c − 1 k +1 n X i =1 φ j + 1 ( λ i ) φ k +1 ( λ i ) c i = δ j k , where w e hav e used the orthog o nal relation Φ C Φ T = C in Lemma 4.1. W e also note that the momen t matrix M ( t ) for the Pfaff lattice has the similar form as the M T oda ( t ) for the indefinite T o da lattice give n in (4.5), M ( t ) = e ξ ( L 2 0 , t ) J e ξ ( L 2 0 , t ) T = Ψ 0 e ξ (Λ 2 , t ) Ψ − 1 0 J Ψ − T 0 e ξ (Λ 2 , t ) Ψ T 0 = Ψ 0 e 2 ξ (Λ 2 , t ) K − 1 J Ψ T 0 =  h ψ 0 i , ψ 0 j e 2 ξ ( z 2 , t ) i  0 ≤ i,j ≤ 2 n − 1 . Comparing with (4 .9), the en tries m ij := h ψ 0 i , ψ 0 j e 2 ξ ( z 2 , t ) i are given b y m ij = − m j i with m 2 j − 1 , 2 k = − m 2 k, 2 j − 1 for j ≤ k , and zero for all other cases, i.e. m 2 j − 1 , 2 k − 1 = m 2 j, 2 k = 0. With this structure of the momen t matrix, o ne can g iv e a direct pro of of Theorem 4.2 without Prop osition 4.2 , ho w ev er this might b e a less elegan t a ppro ac h. Remark 4.3 The skew-inne r pr o duct of (4.14) is closely r elate d to the skew-inner pr o duct for GSE-Pfa ff lattic e, which is define d as h f , g i GSE := n X k =1 { f , g } ( z 2 k ) c k , wher e { f , g } ( z ) = f ′ ( z ) g ( z ) − g ′ ( z ) f ( z ) with f ′ ( z ) = d f ( z ) /d z . This inner pr o duct may b e obtaine d by the fol low ing inner pr o duct in the limit z 2 k − 1 → z 2 k (se e [16]), i.e. h f , g i GSE = lim n X k =1 f ( z 2 k − 1 ) g ( z 2 k ) − f ( z 2 k ) g ( z 2 k − 1 ) z 2 k − z 2 k − 1 c k , In the c ase o f (4.14), we in ste ad take the lim it z 2 k − 1 → − z 2 k . A lso in the c ase of the c ontinuous me as ur e, the inner pr o duct (4.1 4 ) c an b e written by h f , g i = Z Σ [ f ( − z ) g ( z ) − f ( z ) g ( − z )] c ( z 2 ) dz , wher e Σ = R + ∪ i R + oriente d fr om i ∞ to 0 an d then to ∞ . T o r e duc e this expr ession b ack to the d iscr ete c ase we take c ( z 2 ) dz | Σ = n X k =1 δ ( z 2 − z 2 k ) dz    Σ = n X k =1 δ (( z − z k )( z + z k )) dz    Σ = n X k =1  1 2 z k δ ( z − z k ) + 1 2 z k δ ( z + z k )  dz    Σ = n X k =1 1 2 z k δ ( z − z k ) dz    Σ , with z k ∈ Σ , and wher e we use d the p r o p erty δ ( az ) = 1 | a | δ ( z ) for a ∈ R and δ ( az ) = 1 i | a | δ ( z ) for a ∈ i R . The last e quality is b e c ause we have r estricte d to the p ositive and p ositive imagin a ry squar e r o ots with z k ∈ Σ . The r esult is that this d i s c r ete inner pr o duct wil l agr e e with (4.14). 4.4. Asymptotics of the even Pfaff flows In this final s ection, w e men tio n the asymptotic behavior of the ev en Pfaff lattice based on t he results of the indefinite T o da lattice discussed in [17, 18]: T heorem 3.2 implies that the Pfaff flo w on H S -tridiagonal matrix L 0 with c k = ± 1 is equiv alen t to the indefinite T o da flo w on ˜ T = C T giv en b y (3.7). This system was studied in detail in [17] and [18]. It is a v ersion o f the T o da lattice whic h uses HR-factorization in place of QR-factorization, see [1 0 ]. The goal of HR-factorization is to write an elemen t g ∈ S L ( n, R ) as g = r h where r is low er triang ula r and h satisfies hC h T = C . F or g ( t 2 j ) := exp( t 2 j ˜ T ( 0) j ), the HR-fa cto r izat io n g ( t 2 j ) = r ( t 2 j ) h ( t 2 j ) giv es the solution of the t 2 j -flo w of the Pfaff lattice, ˜ T ( t 2 j ) = r − 1 ( t 2 j ) ˜ T ( 0) r ( t 2 j ) = h ( t 2 j ) ˜ T ( 0) h − 1 ( t 2 j ) . Th us the indefinite T o da lattice is a con t inuous v ersion of the HR- algorithm [10]. The indefinite T o da flow may exp erience a blow up, where some of the entries reac h infinit y in finite time [18 ]. A blo w up o ccurs whe n one of the τ functions b ecomes 0. In our case this is pr ecisely when one of the τ 2 k = 0. The initial conditions for a blow-up are c haracterized by Theorem 4.4 (Theorem 3 in [17]) If ˜ T ( 0) p ossess non-r e al e igenvalues or non-r e al eigenve ctors while τ 2 k (0) 6 = 0 , then ˜ T ( t ) blows up to infinity in finite time. The second p ossibilit y o ccurs if the c k do not all hav e the same sign. The case when the eigen v alues of ˜ T are real and the c k do not all ha ve the same sign is studied in Sec tion 4 of [18]. It is sho wn that ev ery flo w (considered in b oth directions t 2 j → ±∞ ) contains a blo w up. W e may then compactify the flow s by adding infinite p oin ts represen ting the blo w ups. It is then sho wn that the fixed p oints of the compactified flo ws are diago nal matrices. In other w or ds a k → 0 and d k → c k z 2 σ ( k ) as t 2 j → ∞ for some p erm utation σ ∈ S n , the symmetric group of order n . When the eigenv alues of ˜ T are real w e may coun t the total n um b er of blo w ups on the flo w. If the num b er of changes of sign in the sequence c k is m then the num b er of blow ups is m ( n − m ) [18]. There is a condition under whic h the indefinite T o da flow, with c k = − 1 for some k , do es not con tain a blo w up in the p o sitiv e t direction. A low er triangular matr ix is called lo w er tr ia ngular tota lly p ositive if all its no n- trivial minor s are p ositiv e. It was sho wn in [12] that if the low er comp onen t of the LU-factorization of the eigenv ector matr ix of ˜ T 0 is lo w er triangular totally p o sitive then there are no blo w ups o n the indefinite T o da flo w for t > 0. If L 0 has only real a nd imaginary eigen v alues, then ˜ T has real eigen v alues. By Theorem 3.2 w e find t ha t the flo w ma y b e con tin ued through a blow up, if it o ccurs, and that this compactified flow con verges : L ( t 2 j ) → diag 2 0 c 1 c 1 z 2 σ (1) 0 ! , . . . , 0 c n c n z 2 σ ( n ) 0 !! , where the eigenv alues of L 0 are {± z m : m = 1 , . . . , n } . If the c k = 1 for all k , then ˜ T is symmetric and no blo w up is p ossible. W e will restrict t o this case from now on. W e app eal to the a bundan t literature on the T o da lattice for the results needed [8, 17, 18]. In particular, under mild assumptions, a s t 2 j → ±∞ , T ( t 2 j ) given b y (3.11 ) con v erges to a diagonal matr ix. As a result w e see that a k ( t 2 j ) → 0. In terms of the Pfaff lattice this is sho wing that L ( t 2 ) con v erges to a 2 × 2 blo c k diagonal matrix a s t 2 → ∞ where the j -th diag onal blo c k has the form 0 1 z 2 j 0 ! . The blo c ks will b e sorted b y the size of z 2 so that the blo c ks with pairs of imaginary eigen v alues will app ear in the low er right corner while those with pairs of real eigen v alues will app ear in the upp er left corner. F or the t 4 -flo w, the eigen v alues will a g ain b e suc h a pair but will now b e s orted b y the size of z 4 , mixing the blo c ks with real and imaginary pairs. While the end result is that the Pfaff lattice restricted to symplectic lo w er Hessen b erg mat rices is equiv alen t in ev ery sense to t he indefinite T o da lattice flow s, this is not a result whic h is immediately obvious from the Pfaff lattice itself. One sees that eve n in the case when the c k = 1 for all k , the long time dynamics of the Pfaff lattice flo ws of symplectic lo w er He ssen b erg matrices is significan tly different from those of non symplectic low er Hessen b erg matrices; for example the the symplectic matrices are in fact fixed p oin ts of the flow. This difference alone merited a separate w ork on these cases and pro duced some interes ting and new connections to generalizations of the T o da la ttice. App endix: SR-algorithm The T oda lattice ma y b e view ed as a contin uous v ersion of the QR -algorithm for diagonalizing a symmetric tridia g onal matrix. In the same w a y the Pfaff lattice is a con tinuous version of the SR- algorithm on an H S -tridiagonal matrix. Here w e collect some p ertinen t details ab out the SR-alg orithm. These results ha v e b een cov ered from a v ariet y of p oin ts of view, a partial list of references is [4, 7, 9, 19, 20 , 2 1, 22, 2 3]. Here w e consider the SR-a lgorithm for a symplectic matrix whic h is symp lectically similar to a low er Hessen b erg matr ix. Recall t ha t an y matrix can b e reduced to a similar Hessen b erg matrix by Householder’s metho d, a fact used in giving an efficien t v ersion of the QR -algorithm. How ev er, note that the Householder’s metho d do es not, in general, giv e a symplectic conjugation. W e will need Theorem 3.4 from [7]: Theorem App endix A.5 L et h ∈ S p ( n, R ) an d s 1 b e a r ow ve ctor in R 2 n . T h en ther e exists a symple ctic tr ansformation S such that S hS − 1 is a lower Hessenb er g symple ctic matrix iff K ( h, s 1 ) has an SR-fa c torization K ( h, s 1 ) = R S wher e R is a lower triangular matrix and s 1 is the first r ow of S ∈ S p ( n, R ) . In ad d ition, if this factorization exists, then L 0 := S hS − 1 = R − 1 C h R is in the lower Hessenb er g form, L 0 =             0 c 1 d 1 0 0 0 a 1 0 · · · 0 2 0 0 a 1 0 0 c 2 d 2 0 · · · 0 2 . . . . . . . . . . . . 0 2 0 2 · · · 0 c n d n 0             , (A.1) that is, L 0 has a 2 × 2 blo ck tridiagon a l form L 0 = ( l i,j ) 1 ≤ i,j ≤ n with 2 × 2 blo ck matric es l i,j having l i,j = 0 2 for | i − j | > 1 and l k ,k = 0 c k d k 0 ! , l k ,k +1 = l k +1 ,k = 0 0 a k 0 ! , wher e c k = ± 1 . A dense set of symplectic matrices h ma y b e placed in this form. W e call a matr ix in the for m (1.6) an “ H S -tridiagonal” mat r ix, whic h play s the similar role as a tridiagonal matrix in t he case of the symmetric matrices. F urthermore, at the expense of excluding a large set of symplectic matrices, w e ma y refine Theorem App endix A.5: Theorem App endix A.6 L et h ∈ S p ( n, R ) , and s 1 b e a r ow ve ctor in R 2 n . Then ther e exists a symple ctic tr ansformation S such that S hS − 1 is a lower Hessenb er g m atrix iff K ( h, s 1 ) h as a n SR-f a ctorization K ( h, s 1 ) = RS wi th R ∈ G k , a n d with s 1 e qual to the first r ow o f S . In add ition, if this factorization exi s ts, then L 0 := S hS − 1 = R − 1 C h R is in an H S -tridiagonal form (1.6) w ith c k = 1 for al l k . W e no w show that if h has quadruples of complex eigen v alues then it will not b e similar to a matrix in the H S -tridiagonal fo rm of (1.6) with c k = 1 for all k : Prop osition App endix A.3 L et L b e a 2 n × 2 n matrix in the form of (1 . 6 ) with c k = 1 for al l k . The e igenvalues of L c ome in r e al or imaginary p airs w ithout multiplicity. Pr o of . One c heck s t ha t for L in this form, w e ha v e L 2 = T ⊗ I 2 where T is a symmetric tridiagonal n × n matrix given b y T =        d 1 a 1 0 · · · 0 a 1 d 2 a 2 · · · 0 . . . . . . . . . . . . . . . 0 0 · · · d n − 1 a n − 1 0 0 · · · a n − 1 d n        (A.2) F rom this o ne computes that F ( λ ) = det  L 2 − λ I 2 n  = [det ( T − λI n )] 2 . As T is symmetric, F ( λ ) only has n real ro ots each with m ultiplicit y 2, therefore the eigen v alues of L are square ro ot s of real nu m b ers, a nd so are only real or imaginary . There is an algo rithm f o r carrying out the transformations of Theorems App endix A.5 and App endix A.6, see [9]. This pro cedure is the analo g ous op eration to the QR- algorithm step of transforming a symmetric matrix to a tridiagonal form by the Householder’s metho d. There is a large time sav ings in carrying out this transformatio n first as an SR-factorizatio n of a matrix in the fo rm of L 0 only requires O ( n ) op erations. The SR-algorit hm is defined as an iterat io n with initial matrix L 0 a symplectic matrix and recursion giv en b y factoring L k − 1 = R k S k using the SR-factorization then taking L k := S k R k = S k L k − 1 S − 1 k = R − 1 k L k − 1 R k . As eac h step is a similarit y transform of the previous step b y b oth a matrix in the symplectic group and a matrix in G k , L k is still a low er Hessen b erg matrix whic h is also symplectic, a nd therefore is still in the H S -tridiagonal fo rm of L 0 ab ov e. If the algo rithm is successful the L k approac hes a family of blo ck diagona l matrices with blo ck s o f the form: (a) 2 × 2 blo c ks containing t w o real eigenv alues ( z , − z ), (b) 2 × 2 blo cks con taining tw o imaginary eigen v alues, ( z , − z ), (c) 4 × 4 blo ck s con ta ining a quadruple of complex eigenv alues ( z , ¯ z , − z , − ¯ z ). In additio n the blo c ks are sorted by the size of ln | z | . If there are complex eigen v alues the sequence do es not con v erge to a fixed matrix, rather it approaches the sorted diagonal shap e. If the c j = 1 for all j , then L k con verges to a blo c k diagonal matrix with j ust 2 × 2 blo c ks. The rate of con v ergence is O ( k 3 ) fo r a dense set of L 0 . In practice one runs the algo r ithm un til sup j | a j ( k ) | at t he k th step is less tha n some fixed ǫ t o lerance. The algorithm also w orks o n the initial matrix h (i.e. without the change t o a low er Hessen b erg L 0 ). There is a substan tial literature on impro ve men ts to this basic algorithm using implicit SR-factorization steps on certain matrix functions of L k rather than just L k (see e.g. [9] and references therein). W e no w sho w that the SR-a lgorithm is directly equiv alent t o the Pfaff lattice flows with a non-traditio na l Hamiltonian. In ligh t of the connection to t he indefinite T o da lattice and HR-algorithms this is not a surprising fact a nd the pro of is appropriat ely close to that for the analo gous fact in the T o da cases. Prop osition App endix A.4 L et L 0 ∈ sp ( n ) in the form of (1.5). The n the SR- algorithm is e q ual to the in te ge r evaluations of the Pfaff lattic e flow with r esp e ct to Hamiltonian H ( L ) = tr ( L ln( L ) − L ) with L (0) = L 0 . Pr o of . Recall that t he SR-algorit hm for the initial matrix L 0 is giv en by L k − 1 = R k S k and L k := S k R k . One can see that L k = S k S k − 1 · · · S 2 S 1 L 0 S − 1 1 S − 1 2 · · · S − 1 k − 1 S − 1 k . While the Pfaff flo ws arise f rom (3.3), L ( t ) = S ( t ) L 0 S ( t ) − 1 where exp ( t ln( L 0 )) = L t 0 = R ( t ) S ( t ) . W e w a n t to show that S ( k ) = S k S k − 1 · · · S 2 S 1 . W e prov e this by induction: First w e c hec k that S (1) = S 1 , fr om whic h L 1 = L (1). Next w e mak e t he inductiv e h yp o thesis that S ( k − 1) = S k − 1 S k − 2 · · · S 2 S 1 and then consider R ( k ) S ( k ) = exp ( k ln( L 0 )) = L k 0 = L k − 1 0 L 0 = R ( k − 1) S ( k − 1) L 0 = R ( k − 1) L k − 1 S ( k − 1) = R ( k − 1) R k S k S ( k − 1) . By uniqueness o f SR - factorizations for R in the iden tity comp onen t of G k , we see tha t S ( k ) = S k S ( k − 1) = S k S k − 1 · · · S 2 S 1 . Ac kno wledgmen ts This work was supp orted b y NSF gr a n t DMS0806219, in additio n V.P . ac knowle dges the partial supp ort of NSF-VIG RE gran t DMS-0 1 35308. References [1] M. Adler, E. 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