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Toda Versus Pfaff Lattice and Related Polynomials

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Toda Versus Pfaff Lattice and Related Polynomials ✐ ✐ 2002/2/7 page 1 ✐ ✐ TODA VERSUS PFAFF LATTICE AND RELATED POLYNOMIALS M. ADLER and P. VAN MOERBEKE Abstract We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilin- ear identities, which we show to be equivalent to the Pfaff Lattice. In the semi-infinite case, the tau-functions are Pfaffians; interesting examples are the matrix integrals over symmetric matrices (symmetric matrix integrals) and matrix integrals over self- dual quaternionic Hermitian matrices (symplectic matrix integrals). There is a striking parallel of the Pfaff lattice with the Toda lattice, and more so, there is a map from one to the other. In particular, we exhibit two maps, dual to each other, (i) from the the Hermitian matrix integrals to the symmetric matrix integrals, and (ii) from the Hermitian matrix integrals to the symplectic matrix integrals. The map is given by the skew-Borel decomposition of a skew-symmetric operator. We give explicit examples for the classical weights. Contents 0. Introduction ............................... 2 1. Splitting theorems, as applied to the Toda and Pfaff lattices ........ 10 2. Wave functions and their bilinear equations for the Pfaff lattice ...... 15 3. Existence of the Pfaff τ-function ..................... 21 4. Semi-infinite matrices m , (skew-)orthogonal polynomials, and matrix in- ∞ tegrals .................................. 30 k 4.1. ∂m /∂tk m , orthogonal polynomials, and Hermitian matrix ∞ = ∞ integrals (α 0) .......................... 30 = k k 4.2. ∂m /∂tk m m ⊤ , skew-orthogonal polynomials, and ∞ = ∞ + ∞ symmetric and symplectic matrix integrals (α 1) ........ 32 =± DUKE MATHEMATICAL JOURNAL Vol. 112, No. 1, c 2002 Received 9 September 1999. Revision received 22 December 2000. 2000 Mathematics Subject Classification. Primary 37K10, 70H06; Secondary 70F45, 82C23. Adler’s work supported by National Science Foundation grant number DMS-98-4-50790. Van Moerbeke’s work supported by National Science Foundation grant number DMS-98-4-50790, a North At- lantic Treaty Organization grant, a Fonds National de la Recherche Scientifique grant, and a Francqui Foun- dation grant. 1 ✐ ✐ ✐ ✐ ✐ ✐ 2002/2/7 page 2 ✐ ✐ 2 ADLER and VAN MOERBEKE 5. A map from the Toda to the Pfaff lattice ................. 37 6. Example 1: From Hermitian to symmetric matrix integrals ........ 41 7. Example 2: From Hermitian to symplectic matrix integrals ........ 46 Appendix A. Free parameter in the skew-Borel decomposition ........ 51 Appendix B. Simultaneous (skew-)symmetrization of L and N ........ 54 Appendix C. Proof of Lemma 3.4 ....................... 55 References .................................. 57 0. Introduction Consider a weight on R, depending on t (t1, t2,...) C , = ∈ ∞ i i ρ (z) g(z) ∞ ti z V (z) ∞ ti z ′ ρt (z) dz e 1 ρ(z) dz e− + 1 dz, with V ′(z) , = = − ρ(z) = = f (z) (0.1) with rational g and f ’s, and with ρ(z) decaying fast enough at . ∞ The Toda lattice, its τ-functions and Hermitian matrix integrals (revisited) This weight leads to a t-dependent moment matrix k ℓ mn(t) µk ℓ(t) z + ρt (z) dz , = + 0 k,ℓ n 1 = R 0 k,ℓ n 1 ≤ ≤ − ≤ ≤ − with the semi-infinite moment matrix m , satisfying the commuting equations ∞ ∂m k k ∞ m m . (0.2) ∂tk = ∞ = ∞ is the customary shift matrix, with zeroes everywhere, except for 1’s just above the diagonal, that is, (v)n vn 1. Consider the Borel decomposition into a lower- and = + an upper-triangular matrix 1 1 m S− S⊤− , (0.3) ∞ = and the following t-dependent matrix integrals (n 0): ≥ i Tr( V (X) ∞ ti X ) τn(t) e − + 1 dX det mn and τ0 1, (0.4) := = = Hn where dX is Haar measure on the ensemble Hn (n n)–Hermitian matrices . As ={ × } is well known (e.g., see E. Witten [18] or M. Adler and P. van Moerbeke [6]), integral (0.4) is a solution to the following two systems. ✐ ✐ ✐ ✐ ✐ ✐ 2002/2/7 page 3 ✐ ✐ TODA VERSUS PFAFF LATTICE 3 (i) The KP-hierarchy∗ 1 ∂2 sk 4(∂) τn τn 0, for k, n 0, 1, 2,.... (0.5) + ˜ − 2 ∂t1∂tk 3 ◦ = = + (ii) The Toda lattice, that is, the tridiagonal matrix 1/2 ∂ τ1 τ0τ2 ∂t log τ 2 0 1 0 τ1 1/2 1/2 τ0τ2 ∂ log τ2 τ1τ3 1 τ 2 ∂t1 τ1 τ 2 L(t) SS− 1 2 , (0.6) := = 1/2 . 0 τ1τ3 ∂ log τ3 .. τ 2 ∂t1 τ2 2 . .. .. satisfies the following commuting Toda equations ∂ L 1 n (L )sk, L , ∂tn = 2 where (A)sk denotes the skew-part of the matrix A for the Lie algebra splitting into skew and lower-triangular matrices. Moreover, the following t-dependent polynomi- als in z, are defined by the S-matrix obtained from the Borel decomposition (0.3); it is also given, on the one hand, in terms of the functions τn(t), and, on the other hand, by a classic determinantal formula (for a C,define a (a, a2/2, a3/3,...) C ) ∈ [ ]:= ∈ ∞ n 1 i n τn(t z− ) pn(t z) Sni(t)z z −[ ] ; := = √τnτn 1 i 0 + = 1 z 1 . det . = τ τ mn(t) . √ n n 1 + n µn,0(t)...µn,n 1(t) z − The pn(t z)’s are orthonormal with respect to the (symmetric) inner-product , sy, ; i j Break/display in footnote defined by z , z sy µij, which is a restatement of the Borel decomposition (0.3) = OK? (see [6]). The vector p(t z) (pn(t z))n 0 is an eigenvector of the matrix L(t) in ; = ; ≥ (0.6): L(t)p(t z) zp(t z). ; = ; t zi i The sℓ’s are the elementary Schur polynomials e 1∞ i si (t)z and sℓ(∂) sℓ(∂/∂t1,(1/2)(∂/∂t2), ∗ := i 0 ˜ := ...).Given a polynomial p(t , t ,...),define the customary Hirota≥ symbol 1 2 p(∂t ) f g p(∂/∂y1,∂/∂y2,...)f (t y)g(t y) . ◦ := + − y 0 = ✐ ✐ ✐ ✐ ✐ ✐ 2002/2/7 page 4 ✐ ✐ 4 ADLER and VAN MOERBEKE The Pfaff lattice and its τ-functions For use throughout this paper, define the skew-symmetric matrix . .. 01 10 0 − 01 2 J 10 , with J I, (0.7) = =− − 01 0 10 − . .. and the involution on the space D gl of infinite matrices, := ∞ J D D a J (a) Ja⊤ J. (0.8) : −→ : −→ := Also, consider the splitting of D k n into two Lie subalgebras k and n, with the = + corresponding projections denoted πk and πn, where k is the Lie algebra of lower- triangular matrices with some special feature (see (1.17)) and where n a D such that Ja⊤ J a sp( ). := { ∈ = }= ∞ Given a skew-symmetric semi-infinite matrix m , consider the commuting differen- ∞ tial equations ∂m i i ∞ m m ⊤ (0.9) ∂ti = ∞ + ∞ ; they maintain the skew-symmetry of m . The Borel decomposition of m into ∞ ∞ lower-triangular times upper-triangular matrices requires the insertion of the skew- symmetric matrix J: 1 1 m (t) Q− (t)JQ− ⊤(t). (0.10) ∞ = Dressing up the shift with the lower-triangular matrix Q(t) leads to the commuting equations (0.11) below. THEOREM 0.1 The Pfaff lattice equations ∂ L i i πk L , L πn L , L (0.11) ∂ti =[− ]=[ ] ✐ ✐ ✐ ✐ ✐ ✐ 2002/2/7 page 5 ✐ ✐ TODA VERSUS PFAFF LATTICE 5 maintain the locus of semi-infinite matrices of the form (ai 0): = 01 d1 a1 O − d1 1 L . (0.12) = d2 a2 − d2 ∗ .. . The solutions L to (0.11) of the form (0.12) are given by 1 L(t) Q(t)Q− (t), = where Q is a lower-triangular matrix, whose entries are given by the coefficients of 1 1 the polynomials, obtained by the finite Taylor expansion in z of τ2n(t z ) below − −[ − ] (h2n τ2n 2(t)/τ2n(t)): := + 2n 1 j 2n 1/2 τ2n(t z− ) q2n(t z) Q2n, j (t)z z h2−n −[ ] , ; := = τ2n(t) j 0 = 2n 1 1 + j 2n 1/2 (z ∂/∂t1)τ2n(t z− ) q2n 1(t z) Q2n 1, j (t)z z h2−n + −[ ] , (0.13) + ; := + = τ2n(t) j 0 = with τ0,τ2,τ4, a sequence of functions of t1, t2,..., characterized by the following bilinear identities for all n, m 0, (τ0 1), ≥ = i dz 1 1 ∞(ti t )z 2n 2m 2 τ2n t z− τ2m 2 t′ z− e 1 − i′ z − − z −[ ] + +[ ] 2πi =∞ i dz ∞(t ti )z 2n 2m τ2n 2 t z τ2m t′ z e 1 i′− − z − 0. (0.14) + + +[ ] −[ ] 2πi = z 0 = Remark Theorem 0.1 is robust and remains valid for the bi-infinite matrix L. In that case, the summations in the expressions q2n and q2n 1 run from j , instead of running + =−∞ from j 0. = The τ-functions are given by Pfaffians pf m2n(t) and satisfy, as a consequence of the bilinear relations (0.14), the Pfaffian KP-hierarchy for k, n 0, 1, 2,..., = 1 ∂2 sk 4(∂) τ2n τ2n sk(∂) τ2n 2 τ2n 2. (0.15) + ˜ − 2 ∂t1∂tk 3 ◦ = ˜ + ◦ − + ✐ ✐ ✐ ✐ ✐ ✐ 2002/2/7 page 6 ✐ ✐ 6 ADLER and VAN MOERBEKE 2 3 The t-dependent polynomials qn(t z) (Q(t)(1, z, z , z ,...) )n in z, obtained in ; = ⊤ (0.13) are “skew-orthonormal” with respect to the skew inner-product , sk,defined i j by y , z sk µij(t), namely, = qi , q j sk 0 i, j< J, ≤ ∞ = and are eigenvectors for the matrix L: L(t)q(t z) zq(t z). (0.16) ; = ; Explicit representations of L, in terms of the τ2n’s, are as follows: L L 00 ˆ 00 ˆ 01 L L L 0 ˆ 10 ˆ 11 ˆ 12 1 1/2 1/2 L QQ− h− L21 L22 L23 h , = = ∗ ˆ ˆ ˆ L32 L33 ..
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