A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy

We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero,...

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Datum:	2013
Hauptverfasser:	Haine, L., Vanderstichelen, D.
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spelling	irk-123456789-1493712019-02-22T01:23:35Z A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy Haine, L. Vanderstichelen, D. We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only ''half of'' a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351]. 2013 Article A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy / L. Haine, D. Vanderstichelen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 17B68 DOI: http://dx.doi.org/10.3842/SIGMA.2013.079 http://dspace.nbuv.gov.ua/handle/123456789/149371 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only ''half of'' a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351].
format	Article
author	Haine, L. Vanderstichelen, D.
spellingShingle	Haine, L. Vanderstichelen, D. A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Haine, L. Vanderstichelen, D.
author_sort	Haine, L.
title	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
title_short	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
title_full	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
title_fullStr	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
title_full_unstemmed	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
title_sort	centerless virasoro algebra of master symmetries for the ablowitz-ladik hierarchy
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149371
citation_txt	A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy / L. Haine, D. Vanderstichelen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 39 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-12T21:58:24Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 079, 42 pages A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz–Ladik Hierarchy? Luc HAINE and Didier VANDERSTICHELEN Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium E-mail: luc.haine@uclouvain.be, didier.vanderstichelen@uclouvain.be Received July 31, 2013, in final form November 30, 2013; Published online December 12, 2013 http://dx.doi.org/10.3842/SIGMA.2013.079 Abstract. We show that the (semi-infinite) Ablowitz–Ladik (AL) hierarchy admits a cen- terless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508–1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Al- gebra Appl. 362 (2003), 29–56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg–de Vries hierarchies which possess only “half of” a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863–911], Damianou [Lett. Math. Phys. 20 (1990), 101–112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329–351]. Key words: Ablowitz–Ladik hierarchy; master symmetries; Virasoro algebra 2010 Mathematics Subject Classification: 37K10; 17B68 Contents 1 Introduction 2 2 Bi-orthogonal Laurent polynomials and CMV matrices 8 2.1 Bi-orthogonal Laurent polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Five term recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Explicit expression for the entries of the CMV matrices . . . . . . . . . . . . . . 13 3 The AL hierarchy and a Lax pair for its master symmetries 15 3.1 The Ablowitz–Ladik hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 A Lax pair for the master symmetries . . . . . . . . . . . . . . . . . . . . . . . . 19 4 The action of the master symmetries on the tau-functions 24 4.1 Some algebraic lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Expression of the master symmetries on the Plücker coordinates . . . . . . . . . 30 4.3 Action of the Virasoro operators L (n) k on the Schur polynomials . . . . . . . . . . 31 4.4 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 References 41 ?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html mailto:luc.haine@uclouvain.be mailto:didier.vanderstichelen@uclouvain.be http://dx.doi.org/10.3842/SIGMA.2013.079 http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 L. Haine and D. Vanderstichelen 1 Introduction The group U(n) of n× n unitary matrices, with Haar measure dU normalized as a probability measure, has eigenvalue probability distribution given by the Weyl formula 1 n! \|∆n(z)\|2 n∏ k=1 dzk 2πizk , zk = eiϕk ∈ S1, ϕk ∈ ]−π, π], with S1 = {z ∈ C : \|z\| = 1} the unit circle, and ∆n(z) the Vandermonde determinant ∆n(z) = det ( zk−1l ) 1≤k,l≤n = ∏ 1≤k<l≤n (zl − zk). (1.1) Thus, for η, θ ∈ ]−π, π], with η ≤ θ, the probability that a randomly chosen matrix from U(n) has no eigenvalues within the arc of circle {z ∈ S1 : η < arg(z) < θ} is given by τn(η, θ) = 1 (2π)nn! ∫ 2π+η θ · · · ∫ 2π+η θ ∏ 1≤k<l≤n \|eiϕk − eiϕl \|2dϕ1 · · · dϕn. Obviously, this probability depends only on the length θ − η. The starting motivation for the present work was our attempt in [24] to understand a diffe- rential equation satisfied by the function −1 2 d dθ log τn(−θ, θ), obtained by Tracy and Widom in [36], from the point of view of the Adler–Shiota–van Moerbeke approach [3], in terms of Virasoro constraints. Introducing the 2-Toda time-dependent tau-functions τn(t, s; η, θ) = 1 n! ∫ [θ,2π+η]n dIn(t, s, z) (1.2) with (t, s) = (t1, t2, . . . , s1, s2, . . .) and dIn(t, s, z) = \|∆n(z)\|2 n∏ k=1 e ∞∑j=1 (tjz j k+sjz −j k ) dzk 2πizk  , deforming the probabilities τn(η, θ) = τn(0, 0; η, θ), we discovered that they satisfy a set of Virasoro constraints indexed by all integers, decoupling into a boundary-part and a time-part 1 i ( eikθ ∂ ∂θ + eikη ∂ ∂η ) τn(t, s; η, θ) = L (n) k τn(t, s; η, θ), k ∈ Z, i = √ −1, (1.3) with the time-dependent operators L (n) k providing a centerless representation of the full Virasoro algebra, that is[ L (n) k , L (n) l ] = (k − l)L(n) k+l, ∀ k, l ∈ Z. (1.4) The basic trick for this result was to use the Lagrangian approach [31] for obtaining Virasoro constraints in matrix models, showing that the following variational formulas hold ∀ k ≥ 0 d dε dIn ( zα 7→ zαe ε(zkα−z −k α ) )∣∣∣ ε=0 = ( L (n) k − L (n) −k ) dIn, d dε dIn ( zα 7→ zαe iε(zkα+z −k α ) )∣∣∣ ε=0 = i ( L (n) k + L (n) −k ) dIn, Master Symmetries of the Ablowitz–Ladik Hierarchy 3 with L (n) k given by L (n) k = k−1∑ j=1 ∂2 ∂tj∂tk−j + n ∂ ∂tk + ∞∑ j=1 jtj ∂ ∂tj+k − ∞∑ j=k+1 jsj ∂ ∂sj−k − k−1∑ j=1 jsj ∂ ∂tk−j − nksk, k ≥ 1, (1.5) L (n) 0 = ∞∑ j=1 jtj ∂ ∂tj − ∞∑ j=1 jsj ∂ ∂sj , (1.6) L (n) −k = − k−1∑ j=1 ∂2 ∂sj∂sk−j − n ∂ ∂sk − ∞∑ j=1 jsj ∂ ∂sj+k + ∞∑ j=k+1 jtj ∂ ∂tj−k + k−1∑ j=1 jtj ∂ ∂sk−j + nktk, k ≥ 1. (1.7) When η = θ, the integral (1.2) is obviously independent of θ, and the left-hand side of (1.3) is equal to zero. By using Weyl’s integration formula, one can recognize it as the partition function of the unitary matrix model, introduced in [30]. After [24] was completed, we found out that our result in this case had already been obtained by Bowick, Morozov and Shevitz [8], though these authors didn’t notice the commutation relations (1.4) of the centerless Virasoro algebra (see Corollary 4.2 and Remark 4.3). Kharchev and Mironov [27] first recognized that the partition function of the unitary matrix model is a special tau function of the two-dimensional Toda lattice (in short 2DTL) hierarchy of Ueno and Takasaki [37], by using bi-orthogonal polynomials on the circle. Then, Kharchev, Mironov and Zhedanov [28, 29] showed that the coefficients entering the Szegö type recursion relations satisfied by these bi-orthogonal polynomials solve the semi-infinite Ablowitz–Ladik (AL in short) hierarchy, a result which is already implicitly contained in [27]. We remind the reader that the first vector field of the AL hierarchy is the system of differential-difference equations introduced by Ablowitz and Ladik [1, 2] in the form ẋn = xn+1 − 2xn + xn−1 − xnyn(xn+1 + xn−1), ẏn = −yn+1 + 2yn − yn−1 + xnyn(yn+1 + yn−1). (1.8) Upon making the change of variable t→ it, when yn = ∓xn the system reduces to the equation −iẋn = xn+1 − 2xn + xn−1 ± \|xn\|2(xn+1 + xn−1), which is a discrete version of the focusing/defocusing nonlinear Schrödinger equation. The functions τn(t, s; η, θ) are thus special instances of tau-functions of the semi-infinite AL hierarchy. The Virasoro constraints they satisfy suggest that the semi-infinite AL hierarchy admits a full centerless Virasoro algebra of additional symmetries (so-called master symmetries), a notion which will be explained below. The goal of this paper is to identify the Virasoro algebra of master symmetries both on the variables xn, yn, n ≥ 0, as well as on the general tau-functions of the AL hierarchy. Since the pioneering works [27, 28, 29] the fact that the semi-infinite AL hierarchy is related to (bi)-orthogonal polynomials on the circle in the same way as the semi-infinite Toda lattice hierarchy is related to orthogonal polynomials on the line, has been rediscovered several times, see for instance [5, 6, 9, 32]. We now introduce the necessary tools to explain this connection. We denote by C [ z, z−1 ] the ring of Laurent polynomials over C. A bilinear form L : C [ z, z−1 ] × C [ z, z−1 ] → C, (f, g) 7→ L[f, g], (1.9) 4 L. Haine and D. Vanderstichelen will be called a bi-moment functional. The bi-moments associated to L are µmn = L [ zm, zn ] , ∀m,n ∈ Z. (1.10) We assume that L satisfies the Toeplitz condition L [ zm, zn ] = L [ zm−n, 1 ] , ∀m,n ∈ Z. (1.11) Because of the Toeplitz condition (1.11), the bi-moments depend only on the difference m− n and we shall often write µmn := µm−n. (1.12) In the rest of the paper, we shall freely use both notations for the bi-moments. An important example of a Toeplitz bi-moment functional is provided by L[f, g] = ∮ S1 f(z)g ( z−1 ) w(z) dz 2πiz , (1.13) with w(z) some weight function on the unit circle S1 which is not necessarily positive or even real valued. We shall also assume L to be quasi-definite, that is det ( µkl ) 0≤k,l≤n−1 6= 0, ∀n ≥ 1. (1.14) This is a necessary and sufficient condition for the existence of a sequence of bi-orthogonal polynomials { p (1) n (z), p (2) n (z) } n≥0 with respect to L, that is p (1) n (z) and p (2) n (z) are polynomials of degree n, satisfying the orthogonality conditions L [ p(1)m (z), p(2)n (z) ] = hnδm,n, hn 6= 0, ∀m,n ∈ N. Introducing the variables xn = p(1)n (0), yn = p(2)n (0), n ≥ 0, (1.15) the monic bi-orthogonal polynomials { p (1) n (z), p (2) n (z) } n≥0 satisfy the Szegö type recurrence re- lations p (1) n+1(z)− zp (1) n (z) = xn+1z np(2)n ( z−1 ) , p (2) n+1(z)− zp (2) n (z) = yn+1z np(1)n ( z−1 ) , (1.16) from which it easily follows that hn+1 hn = 1− xn+1yn+1, n ≥ 0. (1.17) In [5, 6, 27, 28, 29] the AL hierarchy1 is embedded in the 2DTL hierarchy by using a pair (L1, L2) of Hessenberg matrices representing respectively the operator of multiplication C[z] → C[z] : f(z) → zf(z) in the bases p(1)(z) = ( p (1) n (z) ) n≥0 and p(2)(z) = ( p (2) n (z) ) n≥0 of bi-orthogonal polynomials zp(1)(z) = L1p (1)(z), zp(2)(z) = L2p (2)(z). However, to represent the Virasoro algebra of master symmetries, what we shall need is a basis of the ring C [ z, z−1 ] of Laurent polynomials in which both the operators of multiplication by z and z−1 admit nice matrix representations. Thus, we shall adopt the more recent point of view 1In [5, 6] the terminology “Toeplitz hierarchy” instead of “AL hierarchy” is used. Master Symmetries of the Ablowitz–Ladik Hierarchy 5 of Nenciu [32] who used the celebrated Cantero, Moral and Velázquez matrices (CMV matrices in short) to obtain a Lax pair representation for the AL hierarchy in the special defocusing case, that is when yn = xn. We can now describe the content of our paper. To deal with the general AL hierarchy, in Section 2, we first develop a slight generalization of the CMV matrices as introduced in [10]. The generalized CMV matrices are pentadiagonal (semi-infinite) matrices A1, A2 which will represent multiplication by z in bases of bi-orthogonal Laurent polynomials2, which will be denoted by f(z) = (fn(z))n≥0 and g(z) = (gn(z))n≥0, satisfying L[fm, gn] = δm,nhn and the five-term recurrence relations zf(z) = A1f(z), zg(z) = A2g(z). (1.18) In these bases, we shall have that z−1f(z) = A∗1f(z), z−1g(z) = A∗2g(z), with A∗1 = hAT2 h −1, A∗2 = hAT1 h −1 and h the diagonal matrix diag(hn)n≥0, so that A∗1 = A−11 and A∗2 = A−12 . Putting zn = 1− xnyn, with xn and yn defined as in (1.15) (note that x0 = y0 = 1), the matrix A1 reads A1 =  −x1y0 y0 0 −x2z1 −x2y1 −x3 1 z1z2 y1z2 −x3y2 y2 0 O 0 −x4z3 −x4y3 −x5 1 z3z4 y3z4 −x5y4 y4 0 0 ∗ ∗ ∗ 1 O ∗ ∗ ∗ ∗ 0 . . . . . . . . . . . . . . .  , and A2 is obtained from A1 by exchanging the roles of the variables xn and yn. This will be proven at the end of Section 2. To make contact with the work of Nenciu [32] as well as with the authoritative treatises on OPUC by Simon [34, 35], it suffices to specialize to the case xn+1 = −αn, yn+1 = −αn, n ≥ 0, where αn are the so-called Verblunsky coefficients, remembering that x0 = y0 = 1.3 We notice that Gesztesy, Holden, Michor and Teschl [20] have obtained a Lax pair representation for the doubly infinite AL hierarchy, involving a matrix similar to A1 above (up to some conjugation). According to them, the proof is based on “fairly tedious computations”. Our approach via bi-orthogonal Laurent polynomials and the “dressing method” explained below, is more conceptual. In Section 3, we put this theory to use to obtain Lax pair representations both for the AL hierarchy and its Virasoro algebra of master symmetries. Our approach is based on a Favard like theorem which states that there is a one-to-one correspondence between pairs of CMV matrices (A1, A2), with entries built in terms of xn and yn satisfying x0 = y0 = 1 and xnyn 6= 1, n ≥ 1, and quasi-definite Toeplitz bi-moment functionals defined up to a multiplicative nonzero constant. This theorem can be proven as a generalization to bi-orthogonal Laurent polynomials of a similar result in [11], for orthogonal Laurent polynomials on the unit circle. For a complete and independent proof, see [38]. Thus to define the AL hierarchy vector fields Tk, k ∈ Z, it is enough to define them on the bi-moments Tk(µj) ≡ ∂µj ∂tk = µj+k, T−k(µj) ≡ ∂µj ∂sk = µj−k, ∀ k ≥ 1, (1.19) 2The paper [10] considers the case of a sesquilinear hermitian quasi-definite form on C[z, z−1] satisfying the Toeplitz condition, dealing thus with orthogonal instead of bi-orthogonal Laurent polynomials. 3With these notations, the transpose CT of the CMV matrix in [32, 34, 35] is given by CT = ( √ h)−1A1 √ h, with √ h = diag( √ hn)n≥0 and hn+1/hn as in (1.17). 6 L. Haine and D. Vanderstichelen which, in the example of the bi-moment functional (1.13), corresponds to deform the weight w(z) as follows w(z; t, s) = w(z) exp { ∞∑ j=1 ( tjz j + sjz −j)}. (1.20) Obviously [Tk, Tl] = 0, ∀ k, l ∈ Z, if we define T0µj = µj . Then, all the objects introduced above become time dependent. In particular xn(t, s) and yn(t, s) depend on t, s. The Lax pair for the AL hierarchy is then obtained in Theorem 3.4 by “dressing up” the moment equations (1.19) written in matrix form (see (3.10)). Following an idea introduced by Haine and Semengue [23] in the context of the semi-infinite Toda lattice, we define the following vector fields on the bi-moments Vk(µj) = (j + k)µj+k, ∀ k ∈ Z. (1.21) These vector fields trivially satisfy the commutation relations [Vk, Vl] = (l − k)Vk+l, (1.22) [Vk, Tl] = lTk+l, ∀ k, l ∈ Z, (1.23) from which it follows that [[Vk, Tl], Tl] = l[Tk+l, Tl] = 0, ∀ k, l ∈ Z. (1.24) Equations (1.22), (1.23) and (1.24) mean that the vector fields Vk, k ∈ Z, form a centerless Virasoro algebra of master symmetries, in the sense of Fuchssteiner [18], for the AL hierarchy. We remind the reader that master symmetries are generators for time dependent symmetries of the hierarchy which are first degree polynomials in the time variables, that is Xk,l = Vk + t[Vk, Tl], k ∈ Z, are time dependent symmetries of the vector field Tl (run with time t) as one immediately checks that ∂Xk,l ∂t + [Tl, Xk,l] = [Vk, Tl] + [Tl, Vk + t[Vk, Tl]] = 0, from the commutation relations (1.24). Writing (1.21) in matrix form (see (3.18)) and “dressing up” these equations, leads then in Theorem 3.8 to the Lax pair representation of the master symmetries on the CMV matrices (A1, A2), which was our first goal and is a new result. In Section 4, we shall reach our second goal by translating the action of the master symmetries on the tau-functions of the AL hierarchy. One can show (see [5, 28, 29]) that the general solution of the AL hierarchy can be expressed in terms of the Toeplitz determinants τn(t, s) = det ( µk−l(t, s) ) 0≤k,l≤n−1, (1.25) as follows xn(t, s) = Sn(−∂̃t)τn(t, s) τn(t, s) , yn(t, s) = Sn(−∂̃s)τn(t, s) τn(t, s) . In this formula Sn(t), t = (t1, t2, t3, . . .), are the so-called elementary Schur polynomials defined by the generating function exp ( ∞∑ k=1 tkz k ) = ∑ n∈Z Sn(t1, t2, . . .)z n, (1.26) Master Symmetries of the Ablowitz–Ladik Hierarchy 7 and Sn(−∂̃t) = Sn ( − ∂ ∂t1 ,−1 2 ∂ ∂t2 ,−1 3 ∂ ∂t3 , . . . ) , and similarly for Sn(−∂̃s). The functions τn(t, s) are the tau-functions of the semi-infinite AL hierarchy. In the example of the bi-moment func- tional (1.13), a standard computation establishes that τn(t, s) = 1 n! ∫ (S1)n \|∆n(z)\|2 n∏ k=1 w(zk; t, s) dzk 2πizk , (1.27) with w(z; t, s) the deformed weight introduced in (1.20), and ∆n(z) the Vandermonde deter- minant (1.1). Such integrals appear in combinatorics as well as in random matrix theory, see [5, 6, 7, 17, 33, 36] and the references therein. The special case τn(t, s; η, θ) (1.2) con- sidered at the beginning of this Introduction corresponds to w(z) = χ]η,θ[c(z), the characteristic function of the complement of an arc of circle ]η, θ[= {z ∈ S1 : η < arg z < θ}. By a simple computation, which will be recalled in Section 4, one obtains that the tau- functions (1.25) admit the expansion τn(t, s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s), (1.28) where pi0,...,in−1 j0,...,jn−1 = det ( µik−jl(0, 0) ) 0≤k,l≤n−1, (1.29) are the so-called Plücker coordinates, and Si1,...,ik(t) denote the Schur polynomials Si1,...,ik(t) = det ( Sir+s−r(t) ) 1≤r,s≤k. (1.30) In Theorem 4.1, we will show that the induced action of the master symmetries (1.21) on the Plücker coordinates of the tau-function τn(t, s) translates into the centerless Virasoro algebra of partial differential operators L (n) k , k ∈ Z, in the (t, s) variables, that was introduced at the beginning of the Introduction, a result we announced without proof in [24]. For the convenience of the reader, we summarize below our main results, which will be established respectively in Section 3 and Section 4 of the paper. Theorem 1.1. The centerless Virasoro algebra {Vk, k ∈ Z}, of master symmetries of the Ablowitz–Ladik hierarchy which are defined on the bi-moments by (1.21), translates as follows on the CMV matrices and the tau-functions of the hierarchy: 1) On the CMV matrices (A1, A2), the master symmetries admit the Lax pair representation Vk(A1) = [ A1, ( D1A k+1 1 ) −− + ( Ak+1 1 D∗1 ) −− + k ( Ak1 ) −− ] , ∀ k ∈ Z, (1.31) Vk(A2) = [( D2A 1−k 2 ) −− + ( A1−k 2 D∗2 ) −− − k ( A−k2 ) −−, A2 ] , ∀ k ∈ Z, (1.32) where A−− denotes the strictly lower triangular part of A, and D1 and (D∗1)T (respectively D2 and (D∗2)T ) represent the operator of derivation d/dz in the bases (fn(z))n≥0 and ( h−1n gn(z−1) ) n≥0 (respectively (gn(z))n≥0 and ( h−1n fn(z−1) ) n≥0), with fn(z), gn(z) the bi-orthogonal Laurent poly- nomials satisfying (1.18) and L[fm, gn] = hnδm,n. 2) On the tau-functions τn(t, s), the master symmetries are given by a centerless Virasoro algebra of partial differential operators in the (t, s) variables Vkτn(t, s) = L (n) k τn(t, s), ∀ k ∈ Z, with L (n) k defined as in (1.5), (1.6) and (1.7). 8 L. Haine and D. Vanderstichelen 2 Bi-orthogonal Laurent polynomials and CMV matrices In this section, given L : C [ z, z−1 ] × C [ z, z−1 ] → C, a bi-moment functional as in (1.9) which satisfies the Toeplitz condition (1.11) and is quasi-definite (1.14), we construct two sequences of bi-orthogonal Laurent polynomials (in short L-polynomials), which can be thought of as a Gram– Schmidt bi-orthogonalization process applied to the ordered bases { 1, z, z−1, z2, z−2, . . . } and{ 1, z−1, z, z−2, z2, . . . } of C [ z, z−1 ] . They will be called right and left bi-orthogonal L-poly- nomials respectively. This is a slight generalization of the Cantero, Moral and Velázquez [10] construction4. The two sequences of monic right and left bi-orthogonal L-polynomials we shall construct will be expressed in terms of the sequence of monic bi-orthogonal polynomials { p (1) n (z), p (2) n (z) } n≥0, given by the well known formulae p(1)n (z) = 1 τn det  µ0,0 . . . µ0,n−1 1 µ1,0 . . . µ1,n−1 z ... ... ... µn,0 . . . µn,n−1 zn  , p(2)n (z) = 1 τn det  µ0,0 µ0,1 . . . µ0,n ... ... ... µn−1,0 µn−1,1 . . . µn−1,n 1 z . . . zn  , with τn = det ( µkl ) 0≤k,l≤n−1. Denoting by {fn, gn}n≥0 the sequence of monic right bi-orthogonal L-polynomials, multiplication by z in the bases (fn)n≥0 and (gn)n≥0 of C [ z, z−1 ] will be repre- sented by two pentadiagonal matrices A1 and A2, which we call the generalized CMV matrices (and similarly of course for the sequence of left bi-orthogonal L-polynomials). Moreover, the entries of A1 and A2 will have simple expressions in terms of the variables xn and yn entering the Szegö type recurrence relations (1.16). 2.1 Bi-orthogonal Laurent polynomials The following definition is natural from our previous discussion. We define the vector subspaces Lm,n := 〈 zm, zm+1, . . . , zn−1, zn 〉 , ∀m,n ∈ Z, m ≤ n, and for n ≥ 0 L+ 2n := L−n,n, L+ 2n+1 := L−n,n+1, L−2n := L−n,n, L−2n+1 := L−n−1,n, with the convention L+ −1 = L−−1 = {0}. Definition 2.1. A sequence {fn, gn}n≥0 in C[z, z−1] is a sequence of right (left) bi-orthogonal L-polynomials with respect to L if 1) fn, gn ∈ L+(−) n \ L+(−) n−1 ; 2) L[fn, gm] = hnδn,m, with hn 6= 0. 4The paper [10] deals with the case of a sesquilinear quasi-definite hermitian form on C[z, z−1], satisfying the Toeplitz condition. Dropping the condition “hermitian” leads to bi-orthogonal L-polynomials, instead of orthogonal L-polynomials. For the applications we have in mind, see (1.13), it is better to assume L bilinear rather than sesquilinear. Master Symmetries of the Ablowitz–Ladik Hierarchy 9 Remark 2.2. Similarly to orthogonal polynomials, condition (2) in Definition 2.1 can be re- placed equivalently by (3r)  L [ f2n, z k ] = 0, L [ zk, g2n ] = 0 if −n+ 1 ≤ k ≤ n, L [ f2n, z −n] 6= 0, L [ z−n, g2n ] 6= 0, L [ f2n+1, z k ] = 0, L [ zk, g2n+1 ] = 0 if −n ≤ k ≤ n, L [ f2n+1, z n+1 ] 6= 0, L [ zn+1, g2n+1 ] 6= 0, in the case of right bi-orthogonal L-polynomials. For left bi-orthogonal L-polynomials the equiva- lent condition is (3l)  L [ f2n, z k ] = 0, L [ zk, g2n ] = 0 if −n ≤ k ≤ n− 1, L [ f2n, z n ] 6= 0, L [ zn, g2n ] 6= 0, L [ f2n+1, z k ] = 0, L [ zk, g2n+1 ] = 0 if −n ≤ k ≤ n, L [ f2n+1, z −n−1] 6= 0, L [ z−n−1, g2n+1 ] 6= 0. We start by proving that sequences of right and left bi-orthogonal L-polynomials for a given Toeplitz bi-moment functional L are closely related to each other. Proposition 2.3. Let f∗n(z) = fn ( z−1 ) and g∗n(z) = gn ( z−1 ) . Then {fn, gn}n≥0 is a sequence of right bi-orthogonal L-polynomials with respect to L if and only if {g∗n, f∗n}n≥0 is a sequence of left bi-orthogonal L-polynomials with respect to L. Proof. We have f∗n, g ∗ n ∈ L−n \ L−n−1 if and only if fn, gn ∈ L+ n \ L+ n−1. Using the Toeplitz condition (1.11), the result then follows from L [ g∗m(z), f∗n(z) ] = L [ gm ( z−1 ) , fn ( z−1 )] = L [ fn(z), gm(z) ] . � Sequences of right or left bi-orthogonal L-polynomials with respect to L are also very closely related to sequences of bi-orthogonal polynomials for L. This is proven in the next theorem. Theorem 2.4. Let L be a Toeplitz bi-moment functional and let {fn, gn}n≥0 be a sequence in C [ z, z−1 ] . Let us define p (1) 2n (z) = zng2n ( z−1 ) , p (1) 2n+1(z) = znf2n+1(z), p (2) 2n (z) = znf2n ( z−1 ) , p (2) 2n+1(z) = zng2n+1(z). (2.1) The sequence {fn, gn}n≥0 is a sequence of right bi-orthogonal L-polynomials with respect to L if and only if { p (1) n , p (2) n } n≥0 is a sequence of bi-orthogonal polynomials with respect to L. Further- more we have L[fn, gn] = L [ p(1)n , p(2)n ] . (2.2) An analogous statement holds for sequences {fn, gn}n≥0 of left bi-orthogonal L-polynomials, if we define p̃ (1) 2n (z) = znf2n(z), p̃ (1) 2n+1(z) = zng2n+1 ( z−1 ) , p̃ (2) 2n (z) = zng2n(z), p̃ (2) 2n+1(z) = znf2n+1 ( z−1 ) . (2.3) Proof. For n ≥ 0, we define Pn = 〈1, z, . . . , zn〉 the vector subspace of polynomials with degree less than or equal to n, and P−1 := {0}. For { p (1) n , p (2) n } n≥0 defined as in (2.1) it is trivial that p (1) 2n , p (2) 2n ∈ P2n \ P2n−1 ⇔ g2n, f2n ∈ L+ 2n \ L + 2n−1, p (1) 2n+1, p (2) 2n+1 ∈ P2n+1 \ P2n ⇔ f2n+1, g2n+1 ∈ L+ 2n+1 \ L + 2n. 10 L. Haine and D. Vanderstichelen Furthermore we have using the Toeplitz condition (1.11) L [ p (1) 2n+1(z), z k ] = L [ znf2n+1(z), z k ] = L [ f2n+1(z), z k−n], and similarly L [ p (1) 2n (z), zk ] = L [ zn−k, g2n(z) ] , L [ zk, p (2) 2n+1(z) ] = L [ zk−n, g2n+1(z) ] , L [ zk, p (2) 2n (z) ] = L [ f2n(z), zn−k ] . Consequently we have L [ p (1) 2n+1(z), z k ] = 0, 0 ≤ k ≤ 2n ⇔ L [ f2n+1(z), z k ] = 0, − n ≤ k ≤ n, L [ p (1) 2n (z), zk ] = 0, 0 ≤ k ≤ 2n− 1 ⇔ L [ zk, g2n(z) ] = 0, − n+ 1 ≤ k ≤ n, L [ zk, p (2) 2n+1(z) ] = 0, 0 ≤ k ≤ 2n ⇔ L [ zk, g2n+1(z) ] = 0, − n ≤ k ≤ n, L [ zk, p (2) 2n (z) ] = 0, 0 ≤ k ≤ 2n− 1 ⇔ L [ f2n(z), zk ] = 0, − n+ 1 ≤ k ≤ n, and L [ p (1) 2n+1(z), z 2n+1 ] 6= 0 ⇔ L [ f2n+1(z), z n+1 ] 6= 0, L [ p (1) 2n (z), z2n ] 6= 0 ⇔ L [ z−n, g2n(z) ] 6= 0, L [ z2n+1, p (2) 2n+1(z) ] 6= 0 ⇔ L [ zn+1, g2n+1(z) ] 6= 0, L [ z2n, p (2) 2n (z) ] 6= 0 ⇔ L [ f2n(z), z−n ] 6= 0. Thus, according to Remark 2.2, {fn, gn}n≥0 is a sequence of right bi-orthogonal L-polynomials with respect to L if and only if { p (1) n , p (2) n } n≥0 is a sequence of bi-orthogonal polynomials with respect to L. Equation (2.2) follows immediately from the definition (2.1) and the Toeplitz condition (1.11). The statement (2.3) for sequences of left bi-orthogonal L-polynomials is an immediate con- sequence of the result for sequences of right bi-orthogonal L-polynomials and Proposition 2.3. This concludes the proof. � We are now able to prove the existence and the unicity of bi-orthogonal L-polynomials with respect to L. Corollary 2.5. Consider a Toeplitz bi-moment functional L. There exists a sequence of right bi-orthogonal L-polynomials with respect to L and a sequence of left bi-orthogonal L-polynomials with respect to L if and only if L is quasi-definite as defined in (1.14). Each L-polynomial in these sequences is uniquely determined up to an arbitrary non-zero factor. Proof. By virtue of Theorem 2.4, the existence of a sequence of right or left bi-orthogonal L-polynomials with respect to L is equivalent to the existence of a sequence of bi-orthogonal polynomials with respect to L, which are known to exist if and only L is quasi-definite. Since bi-orthogonal polynomials are uniquely determined up to an arbitrary non-zero factor, the same holds for right and left bi-orthogonal L-polynomials. � From now on we shall assume that {fn, gn}n≥0 is a sequence of monic right bi-orthogonal L-polynomials with respect to L, i.e. the coefficients of z−n in f2n, g2n and zn+1 in f2n+1, g2n+1 are equal to 1. We denote by { p (1) n , p (2) n } n≥0 the associated sequence of monic bi-orthogonal polynomials with respect to L, as defined by (2.1). Master Symmetries of the Ablowitz–Ladik Hierarchy 11 2.2 Five term recurrence relations We now prove that bi-orthogonal L-polynomials with respect to a quasi-definite Toeplitz bi- moment functional always satisfy five term recurrence relations. This generalizes the result ob- tained in [10] for orthogonal L-polynomials associated with a quasi-definite Toeplitz sesquilinear hermitian form. The essential ingredient in the proof in [10] is the Toeplitz condition. Conse- quently, it can immediately be translated to the case of bi-orthogonal L-polynomials. Theorem 2.6. Let {fn, gn}n≥0 be a sequence of monic right bi-orthogonal L-polynomials with respect to L, and f∗n(z) = fn ( z−1 ) , g∗n(z) = gn ( z−1 ) . Then for n ≥ 0 there exist five-term recurrence relations zfn(z) = n+2∑ i=n−2 αn,ifi(z), zgn(z) = n+2∑ i=n−2 βn,igi(z), zf∗n(z) = n+2∑ i=n−2 α∗n,if ∗ i (z), zg∗n(z) = n+2∑ i=n−2 β∗n,ig ∗ i (z), where we use the convention fn(z) = gn(z) = 0 if n < 0, and α∗n,i = hn hi βi,n, β∗n,i = hn hi αi,n, with hn = L[fn, gn]. Moreover, we have for all n ≥ 0 α2n−1,2n−3 = 0, α2n,2n+2 = 0, β2n−1,2n−3 = 0, β2n,2n+2 = 0. Proof. As fn ∈ L+ n \L+ n−1, we have zfn(z) ∈ L+ n+2. This implies that zfn admits an expansion in terms of f0, . . . , fn+2 zfn(z) = n+2∑ i=0 αn,ifi(z), with αn,i ∈ C, 0 ≤ i ≤ n+ 2. Consequently, by bi-orthogonality of the sequence {fn, gn}n≥0 we have L[zfn, gm] = n+2∑ i=0 hiαn,iδi,m. But we also have L[zfn, zgk] = L[fn, gk] = 0, 0 ≤ k ≤ n− 1, and 〈g0, . . . , gn−3〉 ⊂ 〈zg0, . . . , zgn−1〉. It follows that L[zfn, gk] = 0, 0 ≤ k ≤ n− 3. Consequently we have αn,i = 0 if i < n− 2, and thus zfn(z) = n+2∑ i=n−2 αn,ifi(z). We prove that α2n,2n+2 = α2n−1,2n−3 = 0. We first prove that α2n,2n+2 = 0. Indeed, we have zf2n(z) ∈ 〈 z1−n, . . . , z1+n 〉 . Consequently, using condition (3r) in Remark 2.2, we have 12 L. Haine and D. Vanderstichelen L[zf2n, g2n+2] = 0 and thus α2n,2n+2 = 0. We also have α2n−1,2n−3 = 0. Indeed, we have L[zf2n−1, g2n−3] = L[f2n−1, z −1g2n−3], and z−1g2n−3(z) ∈ 〈 z1−n, . . . , zn−2 〉 . From condition (3r) in Remark 2.2, it follows that L[zf2n−1, g2n−3] = 0. A similar argument gives β2n,2n+2 = β2n−1,2n−3 = 0. The proof of the other recurrence relations is similar. The coefficients in the recurrence relations satisfy αn,i = L[zfn, gi] L[fi, gi] , βn,i = L[fi, zgn] L[fi, gi] , α∗n,i = L[g∗i , zf ∗ n] L[g∗i , f ∗ i ] , β∗n,i = L[zg∗n, f ∗ i ] L[g∗i , f ∗ i ] . It follows from the definition of {g∗n, f∗n}n≥0 that α∗n,i = L[g∗i , zf ∗ n] L[g∗i , f ∗ i ] = L[fn, zgi] L[fi, gi] = L[fn, zgi] L[fn, gn] L[fn, gn] L[fi, gi] = βi,n hn hi . Similarly we have β∗n,i = L[zg∗n, f ∗ i ] L[g∗i , f ∗ i ] = L[zfi, gn] L[fi, gi] = L[zfi, gn] L[fn, gn] L[fn, gn] L[fi, gi] = αi,n hn hi . This concludes the proof. � Corollary 2.7. With the same notations as in Theorem 2.6 we have z−1fn(z) = n+2∑ i=n−2 α∗n,ifi(z), z−1gn(z) = n+2∑ i=n−2 β∗n,igi(z), z−1f∗n(z) = n+2∑ i=n−2 αn,if ∗ i (z), z−1g∗n(z) = n+2∑ i=n−2 βn,ig ∗ i (z). Defining the vectors f(z) = ( fn(z) ) n≥0, g(z) = ( gn(z) ) n≥0, (2.4) f∗(z) = f ( z−1 ) = ( f∗n(z) ) n≥0, g∗(z) = g ( z−1 ) = ( g∗n(z) ) n≥0, (2.5) the five term recurrence relations obtained in Theorem 2.6 and Corollary 2.7 can be written in vector form zf(z) = A1f(z), zg(z) = A2g(z), z−1f(z) = A∗1f(z), z−1g(z) = A∗2g(z),  zf∗(z) = A∗1f ∗(z), zg∗(z) = A∗2g ∗(z), z−1f∗(z) = A1f ∗(z), z−1g∗(z) = A2g ∗(z), (2.6) with A1 = ( αi,j ) i,j≥0, A2 = ( βi,j ) i,j≥0, where αi,j = βi,j = 0 if \|i− j\| > 2, and A∗1 = hAT2 h −1, A∗2 = hAT1 h −1, (2.7) where h = diag(hn)n≥0. We call the matrices A1, A2 the (generalized) CMV matrices. Clearly, from (2.6), we have A∗1 = A−11 , A∗2 = A−12 . (2.8) Master Symmetries of the Ablowitz–Ladik Hierarchy 13 2.3 Explicit expression for the entries of the CMV matrices Explicit expressions for the entries of the CMV matrices can be found in terms of the variab- les xn, yn introduced in (1.15) entering the Szegö type recurrence relations (1.16). Theorem 2.8. The non-zero entries of the CMV matrices A1 and A2 are (A1)2n−1,2n+1 = 1, (A1)2n−1,2n−1 = −x2ny2n−1, (A1)2n−1,2n = −x2n+1, (A1)2n−1,2n−2 = −x2n(1− x2n−1y2n−1), (A1)2n,2n+1 = y2n, (A1)2n,2n−1 = y2n−1(1− x2ny2n), (A1)2n,2n = −x2n+1y2n, (A1)2n,2n−2 = (1− x2n−1y2n−1)(1− x2ny2n), and (A2)2n−1,2n+1 = 1, (A2)2n−1,2n−1 = −x2n−1y2n, (A2)2n−1,2n = −y2n+1, (A2)2n−1,2n−2 = −y2n(1− x2n−1y2n−1), (A2)2n,2n+1 = x2n, (A2)2n,2n−1 = x2n−1(1− x2ny2n), (A2)2n,2n = −x2ny2n+1, (A2)2n,2n−2 = (1− x2n−1y2n−1)(1− x2ny2n). Proof. (1) We have (A1)2n−1,2n+1 = 1 h2n+1 L [ zf2n−1(z), g2n+1(z) ] . By virtue of Theorem 2.4 we obtain (A1)2n−1,2n+1 = 1 h2n+1 L [ z2−np (1) 2n−1(z), z −np (2) 2n+1(z) ] = 1 h2n+1 L [ z2p (1) 2n−1(z), p (2) 2n+1(z) ] . As z2p (1) 2n−1(z) is a monic polynomial of degree 2n+ 1, using the bi-orthogonality of the polyno- mials, we have (A1)2n−1,2n+1 = 1 h2n+1 L [ z2n+1, p (2) 2n+1(z) ] = 1. (2) We have (A1)2n−1,2n = 1 h2n L[zf2n−1(z), g2n(z)]. By virtue of Theorem 2.4 we obtain (A1)2n−1,2n = 1 h2n L [ z2−np (1) 2n−1(z), z np (1) 2n ( z−1 )] = 1 h2n L [ z2p (1) 2n−1(z), z 2np (1) 2n ( z−1 )] . By using twice (1.16) we have z2p (1) 2n−1(z) = p (1) 2n+1(z)− x2n+1z 2np (2) 2n ( z−1 ) − x2nz2np(2)2n−1 ( z−1 ) , and thus (A1)2n−1,2n = 1 h2n L [ p (1) 2n+1(z), z 2np (1) 2n ( z−1 )] − x2n+1 h2n L [ p (2) 2n (z−1), p (1) 2n ( z−1 )] − x2n h2n L [ p (2) 2n−1 ( z−1 ) , p (1) 2n ( z−1 )] . 14 L. Haine and D. Vanderstichelen As z2np (1) 2n (z−1) is a polynomial of degree 2n, the first term is equal to 0 by bi-orthogonality. The remaining terms give (A1)2n−1,2n = −x2n+1 h2n L [ p (1) 2n (z), p (2) 2n (z) ] − x2n h2n L [ p (1) 2n (z), p (2) 2n−1(z) ] = −x2n+1. (3) We have (A1)2n−1,2n−1 = 1 h2n−1 L[zf2n−1(z), g2n−1(z)]. By virtue of Theorem 2.4 we obtain (A1)2n−1,2n−1 = 1 h2n−1 L [ z2−np (1) 2n−1(z), z 1−np (2) 2n−1(z) ] = 1 h2n−1 L [ zp (1) 2n−1(z), p (2) 2n−1(z) ] . By using (1.16) and then (1.15) we have (A1)2n−1,2n−1 = 1 h2n−1 L [ p (1) 2n (z)− x2nz2n−1p(2)2n−1 ( z−1 ) , p (2) 2n−1(z) ] = − x2n h2n−1 L [ z2n−1p (2) 2n−1 ( z−1 ) , p (2) 2n−1(z) ] = − x2n h2n−1 L [ y2n−1z 2n−1, p (2) 2n−1(z) ] = −x2ny2n−1. (4) We have (A1)2n−1,2n−2 = 1 h2n−2 L[zf2n−1(z), g2n−2(z)]. By virtue of Theorem 2.4 we obtain (A1)2n−1,2n−2 = 1 h2n−2 L [ z2−np (1) 2n−1(z), z n−1p (1) 2n−2 ( z−1 )] = 1 h2n−2 L [ zp (1) 2n−1(z), z 2n−2p (1) 2n−2 ( z−1 )] . Using (1.16) we obtain (A1)2n−1,2n−2 = 1 h2n−2 L [ p (1) 2n (z)− x2nz2n−1p(2)2n−1 ( z−1 ) , z2n−2p (1) 2n−2 ( z−1 )] = 1 h2n−2 L [ p (1) 2n (z), z2n−2p (1) 2n−2 ( z−1 )] − x2n h2n−2 L [ zp (2) 2n−1 ( z−1 ) , p (1) 2n−2 ( z−1 )] . The first term is equal to 0 as z2n−2p (1) 2n−2 ( z−1 ) is a polynomial of degree 2n− 2. Consequently, using (1.17), we have (A1)2n−1,2n−2 = − x2n h2n−2 L [ zp (1) 2n−2(z), p (2) 2n−1(z) ] = − x2n h2n−2 L [ z2n−1, p (2) 2n−1(z) ] = −h2n−1 h2n−2 x2n = −(1− x2n−1y2n−1)x2n. (5) The other relations are proven in a similar way. This finishes the proof. � Master Symmetries of the Ablowitz–Ladik Hierarchy 15 3 The AL hierarchy and a Lax pair for its master symmetries In this section we “dress up” the equations defining the Ablowitz–Ladik hierarchy (1.19) and its master symmetries (1.21) on the bi-moments. This leads to Lax pair representations both for the hierarchy and its master symmetries on the CMV matrices. In all this section we shall denote the time variables (t, s) = (t1, t2, . . . , s1, s2, . . .) of the AL hierarchy by (tk)k∈Z, with t−k = sk, k ≥ 1, and T0 defined as in the Introduction (see below (1.20)). It is only in the next section that the notation (t, s) will be more convenient. 3.1 The Ablowitz–Ladik hierarchy Let χ(z) = ( 1, z, z−1, z2, z−2, . . . )T , (3.1) and let L be a quasi-definite bi-moment functional satisfying the Toeplitz condition. We intro- duce two matrices S1 and S2 by writing the vectors f(z), g(z) (2.4) of monic right bi-orthogonal L-polynomials with respect to L as follows f(z) = S1χ(z), g(z) = h ( ST2 )−1 χ(z), (3.2) with h = diag(hn)n≥0 and hn = L[fn, gn]. With this definition, S1 is a lower triangular matrix with all diagonal elements equal to 1, and S2 is an upper triangular matrix such that h−1S2 has all diagonal elements equal to 1. Associated to L we also define the semi-infinite bi-moment matrix M =  µ0,0 µ0,1 µ0,−1 . . . µ1,0 µ1,1 µ1,−1 . . . µ−1,0 µ−1,1 µ−1,−1 . . . ... ... ... . . .  , (3.3) with µm,n as in (1.10), (1.12). The bi-moment matrix M can be written in terms of the vec- tor χ(z) in (3.1) M = ( L [( χ(z) ) m , ( χ(z) ) n ]) 0≤m,n<∞. The existence of a sequence of right bi-orthogonal L-polynomials for L is equivalent to the existence of a factorisation of the bi-moment matrix M in a product of a lower triangular matrix and an upper triangular matrix with non-zero diagonal elements. Proposition 3.1. The bi-moment matrix M factorizes in a product of a lower triangular matrix and an upper triangular matrix M = S−11 S2. Proof. By bi-orthogonality of the sequence {fn, gn}n≥0, we have L[fm, gn] = hmδm,n. This can be written in matrix form h = ( L[fm, gn] ) 0≤m,n<∞. 16 L. Haine and D. Vanderstichelen Using the expressions (3.2) we obtain h = ( L [( S1χ(z) ) m , ( h ( ST2 )−1 χ(z) ) n ]) 0≤m,n≤∞ = S1MS−12 h. Consequently we have M = S−11 S2, which establishes the result. � We define the semi-infinite shift matrix Λ by Λχ(z) = zχ(z). (3.4) We have Λ =  0 1 0 0 0 0 . . . 0 0 0 1 0 0 . . . 1 0 0 0 0 0 . . . 0 0 0 0 0 1 . . . 0 0 1 0 0 0 . . . ... ... ... ... ... ... . . .  , (3.5) and Λ−1 = ΛT . We leave to the reader to check that, because of the Toeplitz property satisfied by the bi-moments in (3.3), we have the commutation relation [Λ,M ] = 0. (3.6) The CMV matrices can be obtained by “dressing up” the shift Λ. Proposition 3.2. We have A1 = S1ΛS −1 1 , A2 = h ( ST2 )−1 ΛST2 h −1, (3.7) A−11 = S2Λ TS−12 , A−12 = h ( S−11 )T ΛTST1 h −1, (3.8) with S1 and S2 defined in (3.2). Proof. We have A1f(z) = zf(z) = zS1χ(z) = S1Λχ(z) = S1ΛS −1 1 f(z). It follows that A1 = S1ΛS −1 1 . The proof for A2 is similar. The factorisations in (3.8) follow from (2.7), (2.8) and (3.7). � Remember that because L : C[z, z−1] × C[z, z−1] → C is a Toeplitz bi-moment functional, the bi-moments µm,n = L[zm, zn] only depend on the difference m − n and can be written as in (1.12) µm,n := µm−n. The Ablowitz–Ladik hierarchy is defined on the space of bi-moments by the vector fields Tkµj ≡ ∂µj ∂tk = µj+k, ∀ k ∈ Z, (3.9) Master Symmetries of the Ablowitz–Ladik Hierarchy 17 where we have put sk = t−k in (1.19). Obviously, these vector fields satisfy the commutation relations [Tk, Tl] = 0, ∀ k, l ∈ Z. It follows from the definition of Λ in (3.4) and (3.9) that the time evolution of the bi-moment matrix M is given by the equations ∂M ∂tk = ΛkM, ∀ k ∈ Z. (3.10) Equations (3.9) and (3.10) are two equivalent formulations of the Ablowitz–Ladik vector fields at the level of the bi-moments. For a square matrix A, we define • A0 the diagonal part of A; • A− (resp. A+) the lower (resp. upper) triangular part of A; • A−− (resp. A++) the strictly lower (resp. strictly upper) triangular part of A. We establish the following lemma, based on the factorisation of the moment matrix M in Propo- sition 3.1 in a product of a lower triangular and an upper triangular matrix. Lemma 3.3. We have for k ∈ Z ∂S1 ∂tk S−11 = − ( Ak1 ) −−, (3.11) ( ST2 h −1)−1∂(ST2 h−1) ∂tk = ( A−k2 ) −−. (3.12) Proof. On the one hand, we have using Proposition 3.1 ∂M ∂tk = −S−11 ∂S1 ∂tk S−11 S2 + S−11 ∂S2 ∂tk . On the other hand, from equation (3.10) we have ∂M ∂tk = ΛkM = ΛkS−11 S2. As A1 = S1ΛS −1 1 , we obtain Ak1 = −∂S1 ∂tk S−11 + ∂S2 ∂tk S−12 . Since ∂S1 ∂tk is strictly lower triangular, the first term in the right hand side of this equation is strictly lower triangular. The second term is upper triangular. Consequently, taking the strictly lower triangular part of both sides of the equation yields ∂S1 ∂tk S−11 = − ( Ak1 ) −−, which establishes (3.11). To establish the other formula, we write M = ( S−11 h )( h−1S2 ) which gives ∂M ∂tk = ∂ ( S−11 h ) ∂tk ( h−1S2 ) + ( S−11 h )∂(h−1S2) ∂tk . 18 L. Haine and D. Vanderstichelen Using the commutation relation (3.6) and (3.10), we also have ∂M ∂tk = MΛk = ( S−11 h )( h−1S2 ) Λk. As A2 = ( ST2 h −1)−1Λ(ST2 h−1), we obtain after some algebra A−k2 = ∂ ( S−11 h )T ∂tk (( S−11 h )T )−1 + ( ST2 h −1)−1∂(ST2 h−1) ∂tk . Since ( S−11 h )T is upper triangular, the first term in the right hand side of this equation is upper triangular. As ST2 h −1 is lower triangular with all diagonal entries equal to 1, the second term is strictly lower triangular. Consequently, taking the strictly lower triangular part of both sides of the equation yields ( ST2 h −1)−1∂(ST2 h−1) ∂tk = ( A−k2 ) −−, which establishes (3.12), completing the proof. � We are now able to obtain a Lax pair representation for the Ablowitz–Ladik hierarchy. Theorem 3.4. The “dressed up” form of the moment equation (3.10) gives the following Lax pair representation for the Ablowitz–Ladik hierarchy on the semi-infinite CMV matrices (A1, A2) ∂A1 ∂tk = [ A1, ( Ak1 ) −− ] , ∂A2 ∂tk = [ A2, ( A−k2 ) −− ] , ∀ k ∈ Z. (3.13) Proof. As A1 = S1ΛS −1 1 and A2 = ( ST2 h −1)−1Λ(ST2 h−1), we have ∂A1 ∂tk = [ ∂S1 ∂tk S−11 , A1 ] and ∂A2 ∂tk = [ A2, ( ST2 h −1)−1∂(ST2 h−1) ∂tk ] . By Lemma 3.3 we obtain ∂A1 ∂tk = [ − ( Ak1 ) −−, A1 ] and ∂A2 ∂tk = [ A2, ( A−k2 ) −− ] , which establishes (3.13), concluding the proof. � Remark 3.5. Looking back at the explicit expressions for the entries of the CMV matrices in Theorem 2.8, the reader will observe that the entries of A2 are obtained from those of A1 by exchanging the roles of the variables xn and yn. Also A1 contains as entries −x2n+1 and y2n and thus A2 contains as entries x2n and −y2n+1, n ≥ 0 (remember that x0 = y0 = 1). Thus the pair of Lax equations in (3.13) completely determines the Ablowitz–Ladik hierarchy in terms of the variables xn and yn. Using the explicit expressions in terms of the variables xn and yn for the entries of the CMV matrices obtained in Theorem 2.8, and Theorem 3.4, one easily computes the equations for the vector fields T1 and T−1 ∂xn ∂t1 = (1− xnyn)xn+1, ∂xn ∂t−1 = −(1− xnyn)xn−1, ∂yn ∂t1 = −(1− xnyn)yn−1, ∂yn ∂t−1 = (1− xnyn)yn+1. Master Symmetries of the Ablowitz–Ladik Hierarchy 19 After the rescaling xn → e−2txn, yn → e2tyn, the vector field T1−T−1 reduces to the Ablowitz– Ladik equations as written in (1.8). In this paper, we won’t discuss the Hamiltonian structure of the AL hierarchy in terms of the CMV matrices A1 and A2. One can show that for k ≥ 1 ∂xn ∂tk = (1− xnyn) ∂H (1) k ∂yn , ∂xn ∂t−k = (1− xnyn) ∂H (2) k ∂yn , ∂yn ∂tk = −(1− xnyn) ∂H (1) k ∂xn , ∂yn ∂t−k = −(1− xnyn) ∂H (2) k ∂xn , where H (1) k = − 1 k TrAk1, H (2) k = 1 k TrAk2 and Tr denotes the formal trace, see [38] for a proof inspired by [5] in the context of Hessenberg matrices. 3.2 A Lax pair for the master symmetries In this section we translate the action of the master symmetries vector fields Vk, k ∈ Z, defined on the bi-moments by (1.21), on the CMV matrices (A1, A2). We first decompose the vector fields Vk as follows Vk = kTk + Vk, (3.14) where Tk are the Ablowitz–Ladik vector fields (3.9). At the level of the bi-moments, the vector fields Vk are given by Vkµj ≡ d duk µj = jµj+k, j, k ∈ Z. (3.15) These vector fields satisfy the following commutation relations [Vk,Vl] = (l − k)Vk+l, [Vk, Tl] = lTk+l. It follows that [[Vk, Tl], Tl] = 0, ∀ k, l ∈ Z. Consequently, like the vector fields Vk, the vector fields Vk, k ∈ Z, form a Virasoro algebra of master symmetries for the Ablowitz–Ladik hierarchy. The differentiation of χ(z) with respect to z is defined by d dz χ(z) = δχ(z), (3.16) where δ = ∆ΛT , with ∆ = diag(0, 1,−1, 2,−2, . . .), (3.17) and Λ is as in (3.5). Remembering the notation (1.12), (3.15) writes d duk µm,n = (m− n)µm+k,n, which is equivalent to the following equation on the bi-moment matrix M dM duk = ∆ΛkM − ΛkM∆ = [ ∆,ΛkM ] . (3.18) 20 L. Haine and D. Vanderstichelen Remember from (3.2) that f(z) = S1χ(z), g(z) = h ( ST2 )−1 χ(z), (3.19) and, according to (2.6) and (2.7), these vectors satisfy A1f(z) = zf(z), AT1 ( h−1g∗(z) ) = z ( h−1g∗(z) ) , (3.20) A2g(z) = zg(z), AT2 ( h−1f∗(z) ) = z ( h−1f∗(z) ) . (3.21) We define the semi-infinite matrices D1, D ∗ 1 and D2, D ∗ 2 by the relations d dz f(z) = D1f(z), d dz ( h−1g∗(z) ) = (D∗1)T ( h−1g∗(z) ) , (3.22) d dz g(z) = D2g(z), d dz ( h−1f∗(z) ) = (D∗2)T ( h−1f∗(z) ) . (3.23) These matrices can be “dressed up” as explained in the next lemma. Lemma 3.6. We have D1 = S1∆ΛTS−11 , D∗1 = −S2ΛT∆S−12 , (3.24) D2 = ( ST2 h −1)−1∆ΛT ( ST2 h −1), D∗2 = − ( ST1 h −1)−1ΛT∆ ( ST1 h −1), (3.25) with ∆ as in (3.17). Proof. Using (3.19) and (3.22), we have D1f(z) = d dz f(z) = S1 d dz χ(z). By definition of δ in (3.16) and (3.17), we get D1f(z) = S1δS −1 1 f(z) = S1∆ΛTS−11 f(z). This proves the first formula in (3.24). Using (3.19) and remembering from (2.5) that g∗(z) = g ( z−1 ) , we have d dz g∗(z) = h ( ST2 )−1 d dz χ(z−1) = −h ( ST2 )−1 z−2 ( d du χ(u) ) ∣∣∣∣ u=z−1 , which gives, using (3.16), (3.17), (3.19) and remembering the definition (3.4) of the shift mat- rix Λ, d dz g∗(z) = −h ( ST2 )−1 δz−2χ(z−1) = −h ( ST2 )−1 ∆ΛTΛ2χ(z−1) = −h ( ST2 )−1 ∆Λ ( h ( ST2 )−1)−1 g∗(z). Consequently, using the definition (3.22) of D∗1 (D∗1)T ( h−1g∗(z) ) = d dz ( h−1g∗(z) ) = − ( ST2 )−1 ∆ΛST2 ( h−1g∗(z) ) . This proves the second formula in (3.24). The proof of (3.25) is identical to the proof of (3.24) using (3.19) and the definitions of D2 and D∗2 in (3.23). This establishes the lemma. � Master Symmetries of the Ablowitz–Ladik Hierarchy 21 Lemma 3.7. We have for k ∈ Z dS1 duk S−11 = − ( D1A k+1 1 ) −− − ( Ak+1 1 D∗1 ) −−, (3.26) ( ST2 h −1)−1d ( ST2 h −1) duk = − ( D2A 1−k 2 ) −− − ( A1−k 2 D∗2 ) −−. (3.27) Proof. By substituting the factorisation M = S−11 S2 of the moment matrix into (3.18), we obtain −S−11 dS1 duk S−11 S2 + S−11 dS2 duk = ∆ΛkS−11 S2 − ΛkS−11 S2∆. Multiplying this equation on the left by S1 and on the right by S−12 , we get −dS1 duk S−11 + dS2 duk S−12 = S1∆ΛkS−11︸︷︷︸ Term1 −S1ΛkS−11 S2∆S −1 2︸︷︷︸ Term2 . (3.28) Using the factorisation of A1 given in (3.7) and the factorisation of D1 in (3.24), Term1 gives Term1 = S1∆ΛTΛk+1S−11 = ( S1∆ΛTS−11 )( S1Λ k+1S−11 ) = D1A k+1 1 . Similarly, Term2 gives Term2 = Ak1S2∆S −1 2 = Ak+1 1 A−11 S2∆S −1 2 . Using the factorisation of A−11 in (3.8) we get Term2 = Ak+1 1 ( S2Λ TS−12 ) S2∆S −1 2 = Ak+1 1 ( S2Λ T∆S−12 ) = −Ak+1 1 D∗1, where we have used the expression of D∗1 in Lemma 3.6. Substituting these results in (3.28), we obtain −dS1 duk S−11 + dS2 duk S−12 = D1A k+1 1 +Ak+1 1 D∗1. The first term in the left-hand side is strictly lower triangular, while the second term in the left-hand side is upper triangular. Consequently, taking the strictly lower triangular part in both sides, we obtain dS1 duk S−11 = − ( D1A k+1 1 ) −− − ( Ak+1 1 D∗1 ) −−, which establishes (3.26). To establish the other formula, we substitute the factorisation M = ( S−11 h )( h−1S2 ) into equation (3.18) rewritten as dM duk = [ ∆,MΛk ] , which follows from the commutation relation (3.6). This gives d ( S−11 h ) duk ( h−1S2 ) + ( S−11 h )d ( h−1S2 ) duk = ∆ ( S−11 h )( h−1S2 ) Λk − ( S−11 h )( h−1S2 ) Λk∆. 22 L. Haine and D. Vanderstichelen Multiplying this equation on the left by ( S−11 h )−1 and on the right by ( h−1S2 )−1 , we get ( S−11 h )−1d ( S−11 h ) duk + d ( h−1S2 ) duk ( h−1S2 )−1 = ( S−11 h )−1 ∆ ( S−11 h )( h−1S2 ) Λk ( h−1S2 )−1︸︷︷︸ Term1 − ( h−1S2 ) Λk∆ ( h−1S2 )−1︸︷︷︸ Term2 . (3.29) Using the factorisation of A2 in (3.7) and the factorisation of D2 in (3.25), Term2 gives Term2 = ( h−1S2 ) Λk−1Λ∆ ( h−1S2 )−1 = ( h−1S2 ) Λk−1 ( h−1S2 )−1( h−1S2 ) Λ∆ ( h−1S2 )−1 = ( AT2 )1−k DT 2 . Similarly, using the factorisation of A2 in (3.7), gives Term1 = ( S−11 h )−1 ∆ ( S−11 h )( AT2 )−k = ( S−11 h )−1 ∆ ( S−11 h )( AT2 )−1( AT2 )1−k . Using the factorisation of A−12 in (3.8) and the factorisation of D∗2 in (3.25), we get Term1 = ( h−1S1 ) ∆Λ ( h−1S1 )−1( AT2 )1−k = −(D∗2)T ( AT2 )1−k . Substituting these results in the transpose of (3.29), we obtain d ( S−11 h )T duk (( S−11 h )T )−1 + ( ST2 h −1)−1d ( ST2 h −1) duk = −D2A 1−k 2 −A1−k 2 D∗2. Since ( S−11 h )T is upper triangular and ST2 h −1 is lower triangular with diagonal elements equal to 1, by taking the strictly lower part of both sides of this equation, we obtain (3.27). This concludes the proof of the lemma. � We are now able to obtain a Lax pair representation for the master symmetries vector fields Vk, k ∈ Z. Theorem 3.8. The “dressed up” form of the moment equation (3.18) gives the following Lax pair representation for the master symmetries vector fields Vk on the semi-infinite CMV matrices (A1, A2) d duk A1 = [ A1, ( D1A k+1 1 ) −− + ( Ak+1 1 D∗1 ) −− ] , ∀ k ∈ Z, d duk A2 = [( D2A 1−k 2 ) −− + ( A1−k 2 D∗2 ) −−, A2 ] , ∀ k ∈ Z, (3.30) or equivalently d duk A1 = Ak+1 1 + [( D1A k+1 1 ) + − ( Ak+1 1 D∗1 ) −−, A1 ] , ∀ k ∈ Z, d duk A2 = A1−k 2 + [ A2, ( A1−k 2 D∗2)+ − ( D2A 1−k 2 ) −− ] , ∀ k ∈ Z. Proof. As A1 = S1ΛS −1 1 and A2 = ( ST2 h −1)−1Λ(ST2 h−1), we have dA1 duk = [ dS1 duk S−11 , A1 ] and dA2 duk = [ A2, ( ST2 h −1)−1d ( ST2 h −1) duk ] . Using (3.26) and (3.27) in Lemma 3.7, we obtain (3.30). Master Symmetries of the Ablowitz–Ladik Hierarchy 23 From (3.20), (3.22) and from (3.21), (3.23), we deduce that [A1, D1] = 1 and [D∗2, A2] = 1. From these commutation relations, one readily obtains that[ A1, ( D1A k+1 1 ) + ] + [ A1, ( D1A k+1 1 ) −− ] = Ak+1 1 ,[( A1−k 2 D∗2)+, A2 ] + [( A1−k 2 D∗2 ) −−, A2 ] = A1−k 2 , which gives the equivalent formulation for the representation of the master symmetries on the CMV matrices (A1, A2). This concludes the proof. � We notice that as a consequence of the Lax pair representation (3.13) for the AL hierarchy in Theorem 3.4, the relation between the vector fields Vk and Vk in (3.14) and the Lax pair represen- tation (3.30) of Vk in Theorem 3.8, we have established the Lax pair representation (1.31), (1.32) of the vector fields Vk as announced in Theorem 1.1 in the Introduction. We emphasize that Theorem 3.8 exhibits a full centerless Virasoro algebra of master symme- tries for the AL hierarchy. This result stands in contrast with the Toda lattice and Korteweg– de Vries hierarchies which possess only half of a Virasoro algebra of master symmetries Vk, k ≥ −1, satisfying [Vk,Vl] = (l − k)Vk+l, k, l ≥ −1, see [4, 12, 15, 16, 21, 39]. Using the explicit form of the CMV matrices (A1, A2) in Theorem 2.8, and Theorem 3.8, remembering Remark 3.5, one can compute the first few master symmetries vector fields V−2, V−1, V0, V1 in terms of the variables xn, yn: V−2(xn) = (n− 4)xn−2(1− xn−1yn−1)(1− xnyn) − xn−1(1− xnyn) ( (n− 4)xn−1yn + (n− 1)xnyn+1 ) − 2xn−1(1− xnyn) n∑ k=1 ykxk−1 + xn n∑ k=1 y2kx 2 k−1 − 2xn n∑ k=2 ykxk−2 + 2xn n∑ k=2 ykyk−1xk−1xk−2, V−2(yn) = −nyn+2(1− xnyn)(1− xn+1yn+1) + yn+1(1− xnyn) ( nxnyn+1 + (n− 1)xn−1yn ) + 2yn+1(1− xnyn) n∑ k=1 ykxk−1 − yn n∑ k=1 y2kx 2 k−1 + 2yn n∑ k=2 ykxk−2 − 2yn n∑ k=2 ykyk−1xk−1xk−2, V−1(xn) = (n− 2)xn−1(1− xnyn)− xn n∑ k=1 ykxk−1, V−1(yn) = −nyn+1(1− xnyn) + yn n∑ k=1 ykxk−1, V0(xn) = nxn, V0(yn) = −nyn, V1(xn) = nxn+1(1− xnyn)− xn n∑ k=1 xkyk−1, V1(yn) = −(n− 2)yn−1(1− xnyn) + yn n∑ k=1 xkyk−1. 24 L. Haine and D. Vanderstichelen 4 The action of the master symmetries on the tau-functions As we recalled in the Introduction in formula (1.25), the tau-functions of the semi-infinite AL hierarchy are given by τn(t, s) = det ( µk−l(t, s) ) 0≤k,l<n. (4.1) It immediately follows from the generating function of the elementary Schur polynomials (1.26) that ∂ ∂tk Sn(t) = Sn−k(t), (4.2) which shows that the formal solution of the AL hierarchy (1.19) on the moments is µj(t, s) = ∞∑ m,n=0 Sm(t)Sn(s)µj+m−n(0, 0), ∀ j ∈ Z. (4.3) The expansion (1.28) of the tau-functions in terms of the Plücker coordinates (1.29) and the Schur polynomials (1.30) easily follows. Indeed, by substituting (4.3) into (4.1), we have τn(t, s) = ∑ 0≤i0,i1,...,in−1 0≤j0,j1,...,jn−1 det [ µk−l+ik−jl(0, 0) ] 0≤k,l<nSi0(t) · · ·Sin−1(t)Sj0(s) · · ·Sjn−1(s). Relabeling the indices as follows ik 7→ ik − k, jl 7→ jl − l, we get τn(t, s) = ∑ 0≤i0,...,in−1 0≤j0,...,jn−1 det [ µik−jl(0, 0) ] 0≤k,l<nSi0(t)Si1−1(t) · · ·Sin−1−(n−1)(t) × Sj0(s)Sj1−1(s) · · ·Sjn−1−(n−1)(s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 ∑ σ1,σ2∈Sn (−1)σ1(−1)σ2 det [ µik−jl(0, 0) ] 0≤k,l<nSiσ1(0)(t) × Siσ1(1)−1(t) · · ·Siσ1(n−1)−(n−1)(t)Sjσ2(0)(s)Sjσ2(1)−1(s) · · ·Sjσ2(n−1)−(n−1)(s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s), (4.4) with (−1)σ the sign of the permutation σ. The aim of this section is to establish the second part of Theorem 1.1. Theorem 4.1. For all k ∈ Z, we have L (n) k τn(t, s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 Vk ( pi0,...,in−1 j0,...,jn−1 ) Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s), (4.5) with L (n) k , k ∈ Z, defined as in (1.5), (1.6), (1.7), and Vk ( pi0,...,in−1 j0,...,jn−1 ) the Lie derivative of the Plücker coordinates (1.29) in the direction of the master symmetries Vk of the AL hierarchy, as defined in (1.21). This theorem is the key to the quick derivation of the various “Virasoro-type” constraints sa- tisfied by special tau-functions of the AL hierarchy. As an illustration we establish the following result. Master Symmetries of the Ablowitz–Ladik Hierarchy 25 Corollary 4.2. The partition function of the unitary matrix model τn(t, s) = ∫ U(n) exp  ∞∑ j=1 ( tjTrU j + sjTrU−j ) dU, where U(n) is the group of unitary n × n matrices and dU is the standard Haar measure, normalized so that the total volume is 1, satisfies the Virasoro constraints L (n) k τn(t, s) = 0, ∀ k ∈ Z, with L (n) k defined as in (1.5), (1.6) and (1.7). Proof. By using Weyl’s integral formula, one has that τn(t, s) = 1 n! ∫ (S1)n \|∆n(z)\|2 n∏ k=1 exp  ∞∑ j=1 ( tjz j k + sjz −j k ) dzk 2πizk , is a tau-function of the AL hierarchy as in (1.27), with w(z) = 1 in the deformed weight (1.20). Thus the initial moments (at time (t, s) = (0, 0)) are given by µj(0, 0) = ∮ S1 zj dz 2πiz = δj,0, with δj,k the usual Kronecker symbol. By the definition (1.21) of the master symmetries Vk, it follows that Vk(µj)\|(t,s)=(0,0) = (j + k)µj+k(0, 0) = (j + k)δj+k,0 = 0, which, using the definition of the Plücker coordinates (1.29) and formula (4.5), establishes the result. � Remark 4.3. After [24] was completed, we found out that Corollary 4.2, which can be seen as a particular case of our result recalled in (1.3), had already been obtained by Bowick, Morozov and Shevitz [8], using the Lagrangian approach [31] to derive Virasoro constraints. However, these authors didn’t notice the commutation relations (1.4) of the centerless Virasoro algebra. In contrast with Corollary 4.2, the partition function of the Hermitian matrix model (which is a tau- function of the Toda lattice hierarchy) and the partition function of 2d-quantum gravity (which is a tau-function of the KdV hierarchy) are characterized by Virasoro constraints Lkτ(t) = 0, k ≥ −1, corresponding to “half of” a Virasoro algebra, see [4, 14, 19, 22, 25, 30, 31] for the explicit form of the operators Lk in those cases. Actually, in the proof of Theorem 4.1, we shall need to know that the operators L (n) k , k ∈ Z, satisfy the commutation relations of the centerless Virasoro algebra. For the convenience of the reader we repeat the proof given in [24]. Consider the complex Lie algebra A given by the direct sum of two commuting copies of the Heisenberg algebra with bases {~a, aj \| j ∈ Z} and {~b, bj \| j ∈ Z} and defining commutation relations [~a, aj ] = 0, [aj , ak] = jδj,−k~a, [~b, bj ] = 0, [bj , bk] = jδj,−k~b, [~a, ~b] = 0, [aj , bk] = 0, [~a, bj ] = 0, [~b, aj ] = 0, (4.6) 26 L. Haine and D. Vanderstichelen with j, k ∈ Z. Let B be the space of formal power series in the variables t1, t2, . . . and s1, s2, . . . , and consider the following representation of A in B aj = ∂ ∂tj , a−j = jtj , bj = ∂ ∂sj , b−j = jsj , a0 = b0 = µ, ~a = ~b = 1, (4.7) for j > 0, and µ ∈ C. Define the operators A (n) k = 1 2 ∑ j∈Z : a−jaj+k :, B (n) k = 1 2 ∑ j∈Z : b−jbj+k :, (4.8) where k ∈ Z, aj , bj are as in (4.7) with µ = n, and where the colons indicate normal ordering, defined by : ajak := { ajak if j ≤ k, akaj if j > k, and a similar definition for : bjbk :, obtained by changing the a’s in b’s in the former. Expanding the expressions in (4.8) we obtain for k > 0 A (n) 0 = ∑ j>0 jtj ∂ ∂tj + n2 2 , A (n) k = 1 2 ∑ 0<j<k ∂2 ∂tj∂tk−j + ∑ j>k (j − k)tj−k ∂ ∂tj + n ∂ ∂tk , A (n) −k = 1 2 ∑ 0<j<k j(k − j)tjtk−j + ∑ j>k jtj ∂ ∂tj−k + nktk, and similar expressions forB (n) k , by changing the t-variables in s-variables. Using these notations, we can rewrite (1.5), (1.6) and (1.7) as follows L (n) k = A (n) k −B (n) −k + 1 2 k−1∑ j=1 (aj − b−j)(ak−j − bj−k), k ≥ 1, L (n) 0 = A (n) 0 −B(n) 0 , (4.9) L (n) −k = A (n) −k −B (n) k − 1 2 k−1∑ j=1 (a−j − bj)(aj−k − bk−j), k ≥ 1. As shown in [26] (see Lecture 2) the operators A (n) k , k ∈ Z, provide a representation of the Virasoro algebra in B with central charge c = 1, that is[ A (n) k , A (n) l ] = (k − l)A(n) k+l + δk,−l k3 − k 12 , (4.10) for k, l ∈ Z. Similarly, the operators B (n) k satisfy the commutation relations [ B (n) k , B (n) l ] = (k − l)B(n) k+l + δk,−l k3 − k 12 , (4.11) for k, l ∈ Z. Furthermore we have for k, l ∈ Z[ ak, A (n) l ] = kak+l, [ bk, B (n) l ] = kbk+l, [ ak, B (n) l ] = 0, [ bk, A (n) l ] = 0. (4.12) Master Symmetries of the Ablowitz–Ladik Hierarchy 27 Proposition 4.4. The operators L (n) k defined as in (1.5), (1.6), (1.7) satisfy the commutation relations of the centerless Virasoro algebra[ L (n) k , L (n) l ] = (k − l)L(n) k+l, ∀ k, l ∈ Z. (4.13) Proof. We give the proof for k, l ≥ 0, the other cases being similar. As [ A (n) i , B (n) j ] = 0, i, j ∈ Z, we have using (4.6), (4.10), (4.11) and (4.12) [ L (n) k , L (n) l ] = (k − l) ( A (n) k+l −B (n) −k−l ) − 1 2 l−1∑ j=1 j(aj+k − b−j−k)(al−j − bj−l) − 1 2 l−1∑ j=1 (l − j)(aj − b−j)(ak+l−j − bj−k−l) + 1 2 k−1∑ j=1 j(aj+l − b−j−l)(ak−j − bj−k) + 1 2 k−1∑ j=1 (k − j)(aj − b−j)(ak+l−j − bj−k−l). Relabeling the indices in the sums, we have [ L (n) k , L (n) l ] = (k − l) ( A (n) k+l −B (n) −k−l ) − 1 2 k+l−1∑ j=k+1 (j − k)(aj − b−j)(ak+l−j − bj−k−l) − 1 2 l−1∑ j=1 (l − j)(aj − b−j)(ak+l−j − bj−k−l) + 1 2 k+l−1∑ j=l+1 (j − l)(aj − b−j)(ak+l−j − bj−k−l) + 1 2 k−1∑ j=1 (k − j)(aj − b−j)(ak+l−j − bj−k−l) = (k − l)L(n) k+l. This concludes the proof. � The plan of the rest of the section is as follows. After some algebraic preliminaries, we shall translate the master symmetries on the Plücker coordinates pi0,...,in−1 j0,...,jn−1 . Next we shall compute the action of the Virasoro operators on the products Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) of Schur polynomials. Finally we shall end with the proof of Theorem 4.1. 4.1 Some algebraic lemmas We shall need the following lemmas. In order to formulate them, we introduce some notations. Given n vectors x1, . . . , xn ∈ Rn, we shall denote by \|x1x2 . . . xn\| the determinant of the n × n matrix formed with the columns xi. Also, given two vectors x and y, x ∧ y denotes the usual wedge product, with components (x ∧ y)rs = xrys − xsyr. Finally, for an n × n matrix A, Ar will denote the rth column of A, and ATr the rth column of the transposed matrix, and tr(A) will mean the trace of A. With these conventions, we have the following lemma. Lemma 4.5 (Haine–Semengue [23]). Let A and B be n× n matrices, with A invertible. Then (i) n∑ r=1 \|A1 . . . Ar−1BrAr+1 . . . An\| = (detA) tr(BA−1), (ii) ∑ 1≤r<s≤n \|A1 . . . Ar−1BrAr+1 . . . As−1BsAs+1 . . . An\| = (detA) ∑ 1≤r<s≤n (( BA−1 ) r ∧ ( BA−1 ) s ) rs . 28 L. Haine and D. Vanderstichelen Proof. (i) Let A, B be n× n matrices, with A invertible. As A is invertible, its columns form a basis of Cn and thus we have Br = Ac(r) = ∑ j c (r) j Aj , (4.14) for a certain c(r) ∈ Cn, whose components are c (r) j = ( A−1B ) jr . It then follows that n∑ r=1 \|A1 . . . Ar−1BrAr+1 . . . An\| = n∑ r=1 ∣∣∣A1 . . . Ar−1 (∑ j c (r) j Aj ) Ar+1 . . . An ∣∣∣ = detA n∑ r=1 c(r)r = (detA) tr ( BA−1 ) . (ii) Using (4.14), we have∑ 1≤r<s≤n \|A1 . . . Ar−1BrAr+1 . . . As−1BsAs+1 . . . An\| = ∑ 1≤r<s≤n ∣∣∣A1 . . . Ar−1 (∑ j c (r) j Aj ) Ar+1 . . . As−1 (∑ j c (s) j Aj ) As+1 . . . An ∣∣∣ = ∑ 1≤r<s≤n ∣∣∣A1 . . . Ar−1 ( c(r)r Ar + c(r)s As ) Ar+1 . . . As−1 ( c(s)r Ar + c(s)s As ) As+1 . . . An ∣∣∣ = detA ∑ 1≤r<s≤n ( c(r)r c(s)s − c(r)s c(s)r ) = detA ∑ 1≤r<s≤n ( (A−1B)r ∧ (A−1B)s ) rs . We thus obtain∑ 1≤r<s≤n \|A1 . . . Ar−1BrAr+1 . . . As−1BsAs+1 . . . An\| = detA ∑ 1≤r<s≤n (( BA−1 ) r ∧ ( BA−1 ) s ) rs , where we have used the fact that for X, Y two n× n matrices, we have∑ 1≤r<s≤n ( (XY )r ∧ (XY )s ) rs = ∑ 1≤r<s≤n ( (Y X)r ∧ (Y X)s ) rs . (4.15) This concludes the proof of the lemma. � We will also need a transposed version of this lemma. Lemma 4.6. With the same conditions as in Lemma 4.5, we have (i) n∑ r=1 ∣∣AT1 . . . ATr−1(B)Tr A T r+1 . . . A T n ∣∣ = n∑ r=1 \|A1 . . . Ar−1BrAr+1 . . . An\|, (ii) ∑ 1≤r<s≤n ∣∣AT1 . . . ATr−1BT r A T r+1 . . . A T s−1B T s A T s+1 . . . A T n ∣∣ = ∑ 1≤r<s≤n \|A1 . . . Ar−1BrAr+1 . . . As−1BsAs+1 . . . An\|. Proof. Both formulas are direct consequences of Lemma 4.5, by observing that for X, Y two n× n matrices, we have (4.15) and( XT r ∧XT s ) rs = (Xr ∧Xs)rs. � Master Symmetries of the Ablowitz–Ladik Hierarchy 29 We give two consequences of this lemma. First we particularize the preceding lemma to the Plücker coordinates, and then we particularize it to the Schur polynomials. Lemma 4.7. For m ∈ Z we have (i) n∑ l=1 p i0,...,in−1 j0,...,jn−l−m,...,jn−1 = n∑ l=1 pi0,...,in−l+m,...,in−1 j0,...,jn−1 , (ii) ∑ 1≤r<s≤n p i0,...,in−1 j0,...,jn−s−m,...,jn−r−m,...,jn−1 = ∑ 1≤r<s≤n pi0,...,in−s+m,...,in−r+m,...,in−1 j0,...,jn−1 . Proof. Define the n× n matrices A = (µik−jl)0≤k,l≤n−1 and B(m) = (µik−jl+m)0≤k,l≤n−1. We then have n∑ l=1 p i0,...,in−1 j0,...,jn−l−m,...,jn−1 = n∑ l=1 ∣∣A1 . . . An−l−1 ( B(m) ) n−lAn−l+1 . . . An−1 ∣∣ = n∑ l=1 ∣∣AT1 . . . ATn−l−1(B(m) )T n−lA T n−l+1 . . . A T n−1 ∣∣ = n∑ l=1 pi0,...,in−l+m,...,in−1 j0,...,jn−1 , where we have used Lemma 4.6(i) in the second equality. This proves (i). The proof of (ii) is similar. � Lemma 4.8. The following holds (i) n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t) = det  Sin−1−n(t) Sin−1−(n−2)(t) · · · Sin−1(t) Sin−2−n(t) Sin−2−(n−2)(t) · · · Sin−2(t) ... ... ... Si0−n(t) Si0−(n−2)(t) · · · Si0(t)  , (ii) n−1∑ l=1 Sin−1−(n−1),...,in−l−(n−l)+1,...,i1−1(t) = det  Sin−1−(n−1)(t) · · · Sin−1−2(t) Sin−1(t) Sin−2−(n−1)(t) · · · Sin−2−2(t) Sin−2(t) ... ... ... Si1−(n−1)(t) · · · Si1−2(t) Si1(t)  . Proof. We prove (i). Define the n× n matrices A = ( Sin−k−(n−k)+l−k(t) ) 1≤k,l≤n, B(m) = ( Sin−k−(n−k)+l−k+m(t) ) 1≤k,l≤n. We have Sin−1−(n−1),...,i0(t) = detA. It then follows that n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t) = n∑ l=1 ∣∣AT1 . . . ATn−l−1(B(−1) )T n−lA T n−l+1 . . . A T n−1 ∣∣. 30 L. Haine and D. Vanderstichelen Using Lemma 4.6(i) we get n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t) = n∑ l=1 ∣∣A1 . . . An−l−1 ( B(−1) ) n−lAn−l+1 . . . An−1 ∣∣. In the right-hand side, in the lth term, the lth and (l−1)th columns coincide in the determinant, provided that l 6= 1. Consequently, only the first term of the right-hand side gives a non zero contribution. This proves (i). The proof of (ii) is similar. � 4.2 Expression of the master symmetries on the Plücker coordinates We now translate the master symmetries on Plücker coordinates. Lemma 4.9. Let Vkpi0,...,in−1 j0,...,jn−1 denote the Lie derivative of the Plücker coordinates in the direction of the vector fields Vk. Then for k ∈ Z, Vkpi0,...,in−1 j0,...,jn−1 = n−1∑ l=0 (il + k)pi0,...,il−1,il+k,il+1,...,in−1 j0,...,jn−1 − n−1∑ l=0 jlp i0,...,in−1 j0,...,jl−1,jl−k,jl+1,...,jn−1 = n−1∑ l=0 ilpi0,...,il−1,il+k,il+1,...,in−1 j0,...,jn−1 − n−1∑ l=0 (jl − k)p i0,...,in−1 j0,...,jl−1,jl−k,jl+1,...,jn−1 . (4.16) Proof. Fix 0 ≤ i0 < i1 < · · · < in−1 and 0 ≤ j0 < j1 < · · · < jn−1. We introduce the n × n matrices A = ( µik−jl(0, 0) ) 0≤k,l≤n−1, B(m) = ( µik−jl+m(0, 0) ) 0≤k,l≤n−1, as well as the diagonal matrix D = diag(j0, . . . , jn−1). We notice that pi0,...,in−1 j0,...,jn−1 = detA, by definition of the Plücker coordinates. From the definition of Vk and using Leibniz’s rule we find for k ∈ Z Vkpi0,...,in−1 j0,...,jn−1 = n−1∑ l=0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ µi0−j0 µi0−j1 . . . µi0−jn−1 ... ... ... µil−1−j0 µil−1−j1 . . . µil−1−jn−1 (il − j0 + k)µil−j0+k (il − j1 + k)µil−j1+k . . . (il − jn−1 + k)µil−jn−1+k µil−1−j0 µil−1−j1 . . . µil−1−jn−1 ... ... ... µin−1−j0 µin−1−j1 . . . µin−1−jn−1 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , or equivalently, Vkpi0,...,in−1 j0,...,jn−1 = n−1∑ l=0 (il + k)pi0,...,il−1,il+k,il+1,...,in−1 j0,...,jn−1 − n∑ l=1 ∣∣AT1 . . . ATl−1(B(k)D )T l ATl+1 . . . A T n ∣∣. Using Lemma 4.6(i) we obtain Vkpi0,...,in−1 j0,...,jn−1 = n−1∑ l=0 (il + k)pi0,...,il−1,il+k,il+1,...,in−1 j0,...,jn−1 − n∑ l=1 ∣∣A1 . . . Al−1 ( B(k)D ) l Al+1 . . . An ∣∣ = n−1∑ l=0 (il + k)pi0,...,il−1,il+k,il+1,...,in−1 j0,...,jn−1 − n∑ l=1 jl−1 ∣∣A1 . . . Al−1 ( B(k) ) l Al+1 . . . An ∣∣. Master Symmetries of the Ablowitz–Ladik Hierarchy 31 This gives the first equality in (4.16). The second equality in (4.16) can be derived from the first one by using Lemma 4.7(i). � 4.3 Action of the Virasoro operators L (n) k on the Schur polynomials Next we shall compute the action of the Virasoro operators on the products of Schur polynomials Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s). We have the following lemma. Lemma 4.10. (i) L (n) 0 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) = n−1∑ l=0 (il − jl)Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s), (ii) L (n) 1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) = n∑ l=1 in−lSin−1−(n−1),...,in−l−(n−l)−1,...,i0(t)Sjn−1−(n−1),...,j0(s) − n∑ l=1 (jn−l + 1)Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s), (iii) L (n) 2 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) = n∑ l=1 in−lSin−1−(n−1),...,in−l−(n−l)−2,...,i0(t)Sjn−1−(n−1),...,j0(s) + ∑ 1≤k<l≤n Sin−1−(n−1),...,in−k−(n−k)−1,...,in−l−(n−l)−1,...,i0(t)Sjn−1−(n−1),...,j0(s) − n∑ l=1 (jn−l + 2)Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,jn−l−(n−l)+2,...,j0(s) − ∑ 1≤k<l≤n Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,jn−k−(n−k)+1,...,jn−l−(n−l)+1,...,j0(s) + s1 n∑ l=1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) − s1 n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t)Sjn−1−(n−1),...,j0(s). Proof. By using Leibniz’s rule and (4.2) we have for j ≥ 1, ∂ ∂tj Sin−1−(n−1),...,i0(t) = n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−j,...,i0(t), (4.17) and ∂2 ∂t21 Sin−1−(n−1),...,i0(t) = n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−2,...,i0(t) + 2 ∑ 1≤r<s≤n Sin−1−(n−1),...,in−r−(n−r)−1,...,in−s−(n−s)−1,...,i0(t). (4.18) 32 L. Haine and D. Vanderstichelen Define the following n× n matrices A(t) := Sin−1−(n−1)(t) . . . Sin−1(t) ... ... Si0−(n−1)(t) . . . Si0(t)  , B(j, t) := Sin−1−(n−1)−j(t) . . . Sin−1−j(t) ... ... Si0−(n−1)−j(t) . . . Si0−j(t)  , and D = diag(n − 1, n − 2, . . . , 0). We shall denote Â(s) and B̂(j, s) the same matrices with t → s and (i0, . . . , in−1) → (j0, . . . , jn−1). From the definition (1.26) of the elementary Schur polynomials it follows easily that for j ≥ 0, ∞∑ k=1 ktk ∂ ∂tk+j Si(t) = (i− j)Si−j(t), ∞∑ k=j+1 ktk ∂ ∂tk−j Si(t) = (i+ j)Si+j(t)− ∑ 1≤l≤j ltlSi+j−l(t). Consequently, by first using Leibniz’s rule and then Lemma 4.5(i) we have for j ≥ 0 ∞∑ k=1 ktk ∂ ∂tk+j Sin−1−(n−1),...,i0(t) = n∑ l=1 (in−l − j)Sin−1−(n−1),...,in−l−(n−l)−j,...,i0(t) − ( detA(t) ) tr ( A(t)−1B(j, t)D ) , (4.19) ∞∑ k=j+1 ktk ∂ ∂tk−j Sin−1−(n−1),...,i0(t) = n∑ l=1 (in−l + j)Sin−1−(n−1),...,in−l−(n−l)+j,...,i0(t) − ( detA(t) ) tr ( A(t)−1B(−j, t)D ) − j∑ m=1 mtm ( detA(t) ) tr ( A(t)−1B(m− j, t) ) . (4.20) We are now ready to prove the lemma. (i) From (4.9), we have L (n) 0 = A (n) 0 −B(n) 0 . Using (4.19) with j = 0, we obtain A (n) 0 Sin−1−(n−1),...,i0(t) = ∞∑ k=1 ktk ∂ ∂tk Sin−1−(n−1),...,i0(t) + n2 2 Sin−1−(n−1),...,i0(t) = n∑ l=1 in−lSin−1−(n−1),...,i0(t)− ( detA(t) ) tr ( A(t)−1B(0, t)D ) + n2 2 Sin−1−(n−1),...,i0(t). We have B(0, t) = A(t), and thus ( detA(t) ) tr ( A(t)−1B(0, t)D ) = ( detA(t) ) tr(D) = n(n− 1) 2 Sin−1−(n−1),...,i0(t). Consequently, we get A (n) 0 Sin−1−(n−1),...,i0(t) = [ n∑ l=1 in−l + n 2 ] Sin−1−(n−1),...,i0(t). Master Symmetries of the Ablowitz–Ladik Hierarchy 33 Similarly, we get B (n) 0 Sjn−1−(n−1),...,j0(s) = [ n∑ l=1 jn−l + n 2 ] Sjn−1−(n−1),...,j0(s). Combining both equations, we obtain (i). (ii) From (4.9), we have L (n) 1 = A (n) 1 −B(n) −1 . We compute, using (4.17) and (4.19) A (n) 1 Sin−1−(n−1),...,i0(t) = [ ∞∑ j=1 jtj ∂ ∂tj+1 + n ∂ ∂t1 ] Sin−1−(n−1),...,i0(t) = n∑ l=1 ( in−l + n− 1 ) Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t)− ( detA(t) ) tr ( A(t)−1B(1, t)D ) . By virtue of Lemma 4.5(i), we have( detA(t) ) tr ( A(t)−1B(1, t)D ) = (n− 1) ∣∣(B(1, t) ) 1 A2(t) . . . An(t) ∣∣. But by virtue of Lemma 4.8(i), this gives ( detA(t) ) tr ( A(t)−1B(1, t)D ) = (n− 1) n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t). Hence, we obtain A (n) 1 Sin−1−(n−1),...,i0(t) = n∑ l=1 in−lSin−1−(n−1),...,in−l−(n−l)−1,...,i0(t). (4.21) Similarly, we have using (4.20) B (n) −1Sjn−1−(n−1),...,j0(s) =  ∞∑ j=2 jsj ∂ ∂sj−1 + ns1 Sjn−1−(n−1),...,j0(s) = n∑ l=1 (jn−l + 1)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s)− ( det Â(s) ) tr ( Â(s)−1B̂(−1, s)D ) − s1 ( det Â(s) ) tr ( Â(s)−1B̂(0, s) ) + ns1Sjn−1−(n−1),...,j0(s). We have using Lemma 4.5(i)( det Â(s) ) tr ( Â(s)−1B̂(−1, s)D ) = 0, and, obviously, we also have( det Â(s) ) tr ( Â(s)−1B̂(0, s) ) = nSjn−1−(n−1),...,j0(s). Consequently we obtain B (n) −1Sjn−1−(n−1),...,j0(s) = n∑ l=1 (jn−l + 1)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s). (4.22) Subtracting (4.21) and (4.22) gives (ii). 34 L. Haine and D. Vanderstichelen (iii) From (4.9), we have L (n) 2 = A (n) 2 −B(n) −2 + 1 2 ( ∂ ∂t1 − s1 )2 . We study separately the contributions of the three terms in the operator L (n) 2 on the product of Schur functions. We start with the contribution of A (n) 2 . We compute, using (4.17), (4.18) and (4.19) A (n) 2 Sin−1−(n−1),...,i0(t) = 1 2 ∂2 ∂t21 + ∞∑ j=1 jtj ∂ ∂tj+2 + n ∂ ∂t2 Sin−1−(n−1),...,i0(t) = n∑ l=1 ( in−l + n− 3 2 ) Sin−1−(n−1),...,in−l−(n−l)−2,...,i0(t) + ∑ 1≤k<l≤n Sin−1−(n−1),...,in−k−(n−k)−1,...,in−l−(n−l)−1,...,i0(t) − ( detA(t) ) tr ( A(t)−1B(2, t)D ) . The last term in this equation gives by developing the trace( detA(t) ) tr ( A(t)−1B(2, t)D ) = ( detA(t) )[ (n− 1) ( A(t)−1B(2, t) ) 11 + (n− 2) ( A(t)−1B(2, t) ) 22 ] . We have( detA(t) ) tr ( A(t)−1B(2, t) ) = ( detA(t) )[( A(t)−1B(2, t) ) 11 + ( A(t)−1B(2, t) ) 22 ] , and by a short computation( A(t)−1B(2, t) ) 22 = − ∑ 1≤k<l≤n (( A(t)−1B(1, t) ) k ∧ ( A(t)−1B(1, t) ) l ) kl . Consequently we have( detA(t) ) tr ( A(t)−1B(2, t)D ) = (n− 1) ( detA(t) ) tr ( A(t)−1B(2, t) ) + ( detA(t) ) ∑ 1≤k<l≤n (( A(t)−1B(1, t) ) k ∧ ( A(t)−1B(1, t) ) l ) kl . Using Lemma 4.5, we obtain ( detA(t) ) tr ( A(t)−1B(2, t)D ) = (n− 1) n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−2,...,i0(t) + ∑ 1≤k<l≤n Sin−1−(n−1),...,in−k−(n−k)−1,...,in−l−(n−l)−1,...,i0(t). Hence, we get A (n) 2 Sin−1−(n−1),...,i0(t) = n∑ l=1 ( in−l − 1 2 ) Sin−1−(n−1),...,in−l−(n−l)−2,...,i0(t). (4.23) Master Symmetries of the Ablowitz–Ladik Hierarchy 35 We now turn to the contribution of B (n) −2 . We have using (4.20) B (n) −2Sjn−1−(n−1),...,j0(s) = 1 2 s21 + ∞∑ j=3 jsj ∂ ∂sj−2 + 2ns2 Sjn−1−(n−1),...,j0(s) = [ 1 2 s21 + 2ns2 ] Sjn−1−(n−1),...,j0(s) + n∑ l=1 ( jn−l + 2 ) Sjn−1−(n−1),...,jn−l−(n−l)+2,...,j0(s) − ( det Â(s) ) tr ( Â(s)−1B̂(−2, s)D ) − 2∑ m=1 msm ( det Â(s) ) tr ( Â(s)−1B̂(m− 2, s) ) . By a similar argument as above, we have( det Â(s) ) tr ( Â(s)−1B̂(−2, s)D ) = − ( det Â(s) ) ∑ 1≤k<l≤n (( Â(s)−1B̂(−1, s) ) k ∧ ( Â(s)−1B̂(−1, s) ) l ) kl , and thus using Lemma 4.5(ii), we obtain( det Â(s) ) tr ( Â(s)−1B̂(−2, s)D ) = − ∑ 1≤k<l≤n Sjn−1−(n−1),...,jn−k−(n−k)+1,...,jn−l−(n−l)+1,...,j0(s). We also have, using Lemma 4.5(i), 2∑ m=1 msm ( det Â(s) ) tr ( Â(s)−1B̂(m− 2, s) ) = s1 n∑ l=1 Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) + 2ns2Sjn−1−(n−1),...,j0(s). Consequently, we have B (n) −2Sjn−1−(n−1),...,j0(s) = n∑ l=1 ( jn−l + 2 ) Sjn−1−(n−1),...,jn−l−(n−l)+2,...,j0(s) + ∑ 1≤k<l≤n Sjn−1−(n−1),...,jn−k−(n−k)+1,...,jn−l−(n−l)+1,...,j0(s) − s1 n∑ l=1 Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) + 1 2 s21Sjn−1−(n−1),...,j0(s). (4.24) Finally, we turn to the contribution of the term 1 2 ( ∂ ∂t1 −s1 )2 . We have using (4.17) and (4.18) 1 2 [ ∂ ∂t1 − s1 ]2 Sin−1−(n−1),...,i0(t) = 1 2 [ ∂2 ∂t21 − 2s1 ∂ ∂t1 + s21 ] Sin−1−(n−1),...,i0(t) = 1 2 n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−2,...,i0(t) + ∑ 1≤k<l≤n Sin−1−(n−1),...,in−k−(n−k)−1,...,in−l−(n−l)−1,...,i0(t) − s1 n∑ l=1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t) + 1 2 s21Sin−1−(n−1),...,i0(t). (4.25) Combining (4.23), (4.24) and (4.25), we obtain (iii). � 36 L. Haine and D. Vanderstichelen Remark 4.11. We observe that by definition of the operators L (n) k we have L (n) −kSin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) = −L(n) k Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j0(s) ∣∣∣ t↔s (i0,...,in−1)↔(j0,...,jn−1) . 4.4 Proof of the main theorem We now turn to the last part of this section. We will prove Theorem 4.1. We first prove the following lemma. Lemma 4.12. n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) + ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Sin−1−(n−1),...,i0(t)Sjn−1−(n−1),...,j1−1,0(s) = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t)Sjn−1−(n−1),...,j0(s). (4.26) Proof. For simplicity, we will use the notations Si(t) = Sin−1−(n−1),...,i0(t), Sj(s) = Sjn−1−(n−1),...,j0(s), (4.27) when no ’special’ shift on the indices of the Schur functions occur. Relabeling each term in the first sum of the left-hand side of (4.26) in the following way jn−l 7→ jn−l − 1 gives n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Si(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−l−1<···<jn−1 p i0,...,in−1 j0,...,jn−l−1,...,jn−1 Si(t)Sj(s). On the one hand, for a fixed 1 ≤ l ≤ n− 1, if jn−l = jn−l−1 + 1, then p i0,...,in−1 j0,...,jn−l−1,...,jn−1 = 0. On the other hand, for a fixed 2 ≤ l ≤ n, if jn−l = jn−l+1, then Sjn−1−(n−1),...,jn−l−(n−l),...,j0(s) = 0. Therefore n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Si(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 p i0,...,in−1 j0,...,jn−l−1,...,jn−1 Si(t)Sj(s) − ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s). Master Symmetries of the Ablowitz–Ladik Hierarchy 37 Consequently, the left-hand side of (4.26) is equal to n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Si(t)Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j0(s) + ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s) = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 p i0,...,in−1 j0,...,jn−l−1,...,jn−1 Si(t)Sj(s). (4.28) Similarly, one can show that the right-hand side of (4.26) is equal to n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 Sin−1−(n−1),...,in−l−(n−l)−1,...,i0(t)Sj(s) = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−l+1,...,in−1 j0,...,jn−1 Si(t)Sj(s). (4.29) By virtue of Lemma 4.7(i), (4.28) and (4.29) are equal. � Proof of Theorem 4.1. We will prove the theorem for k ≥ 0. The case k < 0 is similar. Using the Plücker expansion (4.4) of τn(t), and Lemmas 4.9 and 4.10 we have for k = 0, 1, using the notations (4.27), Vkτn(s, t) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 Vkpi0,...,in−1 j0,...,jn−1 Si(t)Sj(s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 L (n) k Si(t)Sj(s) = L (n) k τn(s, t), where, in the second equality, we have performed some relabeling of the indices as in the proof of Lemma 4.12. We will finish the proof with the case k = 2, for which we provide some more details, but first we prove the theorem for general k ≥ 3. We proceed by induction. Assume the theorem holds for some k ≥ 2. We will establish it for k + 1. The argument follows from the commutation relations (4.13) and (1.22). We have (k − 1)Vk+1τn(s, t) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 [V1, Vk]pi0,...,in−1 j0,...,jn−1 Si(t)Sj(s) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 [ L (n) k , L (n) 1 ] Si(t)Sj(s) = (k − 1)L (n) k+1τn(s, t), where in the second equality we have used the induction hypothesis. We now provide some details for the case k = 2. Using Lemmas 4.10 and 4.12 we have L (n) 2 τn(s, t) = T1 + T2 + T3 + T4 − s1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s), (4.30) 38 L. Haine and D. Vanderstichelen with T1 := ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 n∑ l=1 in−lSin−1−(n−1),...,in−l−(n−l)−2,...,i0(t)Sj(s), T2 := − ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 n∑ l=1 (jn−l + 2)Si(t)Sjn−1−(n−1),...,jn−l−(n−l)+2,...,j0(s), T3 := ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 ∑ 1≤k<l≤n Sin−1−(n−1),...,in−k−(n−k)−1,...,in−l−(n−l)−1,...,i0(t)Sj(s), T4 := − ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−1 j0,...,jn−1 × ∑ 1≤k<l≤n Si(t)Sjn−1−(n−1),...,jn−k−(n−k)+1,...,jn−l−(n−l)+1,...,j0(s). We will consider separately the four terms T1, T2, T3, T4. By arguments similar to those used in the proof of Lemma 4.12, and using the fact that Sin−1−(n−1),...,i0(t) = 0 if ik < 0 for some 0 ≤ k ≤ n− 1, we get for T1 T1 = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s) + n−1∑ l=1 ∑ −1≤i0−1<···<in−l−1−1 =in−l<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s) − n∑ l=2 ∑ −1≤i0−1<···<in−l−1−1 <in−l+1=in−l+1<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s). The two last terms in this expression annihilate, i.e. 0 = n−1∑ l=1 ∑ −1≤i0−1<···<in−l−1−1 =in−l<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s) − n∑ l=2 ∑ −1≤i0−1<···<in−l−1−1 <in−l+1=in−l+1<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s). (4.31) Indeed, we have for 1 ≤ l ≤ n− 1∑ −1≤i0−1<···<in−l−1−1 =in−l<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s) = ∑ −1≤k0−1<···<kn−l−1 =kn−l−1<···<kn−1 0≤j0<···<jn−1 (kn−l−1 + 2)pk0,...,kn−l−2,kn−l,kn−l−1+2,kn−l+1,...,kn−1 j0,...,jn−1 × Skn−1−(n−1),...,kn−l+1−(n−l+1),kn−l−1−(n−l),kn−l−(n−l−1),kn−l−2−(n−l−2),...,k0(t)Sj(s), Master Symmetries of the Ablowitz–Ladik Hierarchy 39 where we have made the relabeling in−l−1 7→ kn−l, in−l 7→ kn−l−1, and im 7→ km if m 6= n− l− 1, n− l. As the Plücker coordinates and the Schur functions are determinants, we have, permuting lines in the determinants, pk0,...,kn−l−2,kn−l,kn−l−1+2,kn−l+1,...,kn−1 j0,...,jn−1 = −pk0,...,kn−l−1+2,...,kn−1 j0,...,jn−1 , and Skn−1−(n−1),...,kn−l+1−(n−l+1),kn−l−1−(n−l),kn−l−(n−l−1),kn−l−2−(n−l−2),...,k0(t) = −Sk(t), and hence ∑ −1≤i0−1<···<in−l−1−1=in−l<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s) = ∑ −1≤k0−1<···<kn−l−1=kn−l−1<···<kn−1 0≤j0<···<jn−1 (kn−l−1 + 2)pk0,...,kn−l−1+2,...,kn−1 j0,...,jn−1 Sk(t)Sj(s). Summing this expression for 1 ≤ l ≤ n− 1, and relabeling l 7→ l− 1 we get (4.31). Consequently we obtain T1 = n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 (in−l + 2)pi0,...,in−l+2,...,in−1 j0,...,jn−1 Si(t)Sj(s). (4.32) By similar arguments, we have T2 = − n∑ l=1 ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 jn−lp i0,...,in−1 j0,...,jn−l−2,...,jn−1 Si(t)Sj(s) + ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,1(s), (4.33) T3 = ∑ 1≤k<l≤n ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 pi0,...,in−l+1,...,in−k+1,...,in−1 j0,...,jn−1 Si(t)Sj(s), (4.34) T4 = − ∑ 1≤k<l≤n ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 p i0,...,in−1 j0,...,jn−l−1,...,jn−k−1,...,jn−1 Si(t)Sj(s) + ∑ 1≤k≤n−1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−k−1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s). (4.35) Substituting (4.32), (4.33), (4.34) and (4.35) in (4.30), using Lemma 4.7(ii) and Lemma 4.9 we obtain L (n) 2 τn(s, t) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 V2pi0,...,in−1 j0,...,jn−1 Si(t)Sj(s) + n−1∑ k=1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−k−1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s) 40 L. Haine and D. Vanderstichelen − s1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s) + ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,1(s). We prove that the last three terms in this expression annihilate 0 = n−1∑ k=1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−k−1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s) − s1 ∑ 0≤i0<···<in−1 0<j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,0(s) + ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,1(s), (4.36) and hence L (n) 2 τn(s, t) = ∑ 0≤i0<···<in−1 0≤j0<···<jn−1 V2pi0,...,in−1 j0,...,jn−1 Si(t)Sj(s). (4.37) Indeed, developing the determinant Sjn−1−(n−1),...,j1−1,1(s) with respect to the last line, using the fact that the first elementary Schur polynomials are S0(s) = 1 and S1(s) = s1, and Lemma 4.8(ii), we have∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,1(s) = s1 ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1(s) − ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t) n−1∑ l=1 Sjn−1−(n−1),...,jn−l−(n−l)+1,...,j1−1(s). By an argument similar to that of the proof of Lemma 4.12, we get∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1,1(s) = s1 ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1(s) − n−1∑ l=1 ∑ 0≤i0<···<in−1 0≤j1<···<jn−1 p i0,...,in−1 −1,j1,...,jn−l−1,...,jn−1 Si(t)Sjn−1−(n−1),...,j1−1(s). 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A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy

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