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, Moral and Vel\'azquez (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].

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 differential equation satisfied by the function − 1 2 d dθ log τ n (−θ, θ), obtained by Tracy and Widom in [34], 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 τ n (t, s; η, θ) = 1 n! [θ,2π+η] n dI n (t, s, z) (1. 2) with (t, s) = (t 1 , t 2 , . . . , s 1 , s 2 , . . .) and dI n (t, s, z) = |∆ n (z)| 2 n k=1 e ∞ j=1 (t j z j k +s j z −j k ) dz k 2πiz k , deforming the probabilities τ n (η, θ) = τ n (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 e ikθ ∂ ∂θ + e ikη ∂ ∂η τ n (t, s; η, θ) = L The basic trick for this result was to use the Lagrangian approach [29] for obtaining Virasoro constraints in matrix models, showing that the following variational formulas hold ∀k ≥ 0 d dε dI n z α → z α e ε(z k with L (n) k given by The functions τ n (t, s; η, θ) are special instances of tau-functions of the semi-infinite Ablowitz-Ladik (AL in short) 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. We remind the reader that the first vector field of the AL hierarchy is the system of differentialdifference equations introduced by Ablowitz and Ladik [1,2] in the forṁ x n = x n+1 − 2x n + x n−1 − x n y n (x n+1 + x n−1 ), y n = −y n+1 + 2y n − y n−1 + x n y n (y n+1 + y n−1 ). (1.8) Upon making the change of variable t → it, when y n = ∓ x n the system reduces to the equation which is a discrete version of the focusing/defocusing nonlinear Schrödinger equation. The goal of this paper is to identify the Virasoro algebra of master symmetries both on the variables x n , y n , n ≥ 0, as well as on the general taufunctions of the AL hierarchy. As discovered in recent years [5,6,9,30], 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. 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 (1.9) will be called a bi-moment functional. The bi-moments associated to L are µ mn = L[z m , z n ], ∀m, n ∈ Z.
(1. 10) We assume that L satisfies the Toeplitz condition L[z m , z n ] = L[z m−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 µ mn := µ m−n .
(1. 12) In the rest of the paper, we shall freely use both notations for the bimoments. An important example of a Toeplitz bi-moment functional is provided by , (1.13) with w(z) some weight function on the unit circle S 1 which is not necessarily positive or even real valued. We shall also assume L to be quasi-definite, that is det µ kl 0≤k,l≤n−1 = 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)] = h n δ m,n , h n = 0, ∀m, n ∈ N.
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 of Nenciu [30] 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 y n = x n . 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 A 1 , A 2 which will represent multiplication by z in bases of bi-orthogonal Laurent polynomials 2 , which will be denoted by f (z) = (f n (z)) n≥0 and g(z) = (g n (z)) n≥0 , satisfying L(f m , g n ) = δ m,n h n and the five-term recurrence relations In these bases, we shall have that Putting z n = 1 − x n y n , with x n and y n defined as in (1.15) (note that x 0 = y 0 = 1), the matrix A 1 reads and A 2 is obtained from A 1 by exchanging the roles of the variables x n and y n . This will be proven at the end of Section 2. To make contact with the work of Nenciu [30] as well as with the authoritative treatises on OPUC by Simon [32,33], it suffices to specialize to the case x n+1 = −α n , y n+1 = −α n , n ≥ 0, where α n are the so-called Verblunsky coefficients, remembering that x 0 = y 0 = 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 A 1 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 (A 1 , A 2 ), with entries built in terms of x n and y n satisfying x 0 = y 0 = 1 and x n y n = 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 [35]. Thus to define the AL hierarchy vector fields T k , k ∈ Z, it is enough to define them on the bi-moments which, in the example of the bi-moment functional (1.13), corresponds to deform the weight w(z) as follows Obviously [T k , T l ] = 0, ∀k, l ∈ Z, if we define T 0 µ j = µ j . Then, all the objects introduced above become time dependent. In particular x n (t, s) and y n (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 (1.21) These vector fields trivially satisfy the commutation relations from which it follows that Equations (1.22), (1.23) and (1.24) mean that the vector fields V k , k ∈ Z, form a centerless Virasoro algebra of master symmetries, in the sense of 3 With these notations, the transpose C T of the CMV matrix in [30,32,33] is given by 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 are time dependent symmetries of the vector field T l (run with time t) as one immediately checks that 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 (A 1 , A 2 ), 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]) that the general solution of the AL hierarchy can be expressed in terms of the Toeplitz determinants as follows x n (t, s) = S n (−∂ t )τ n (t, s) τ n (t, s) , y n (t, s) = S n (−∂ s )τ n (t, s) τ n (t, s) .
In this formula S n (t), t = (t 1 , t 2 , t 3 , . . .), are the so-called elementary Schur polynomials defined by the generating function . . , and similarly for S n (−∂ s ). The functions τ n (t, s) are the tau-functions of the semi-infinite AL hierarchy. In the example of the bi-moment functional (1.13), a standard computation establishes that with w(z; t, s) the deformed weight introduced in (1.20), and ∆ n (z) the Vandermonde determinant (1.1). Such integrals appear in combinatorics as well as in random matrix theory, see [5,6,7,17,31,34] and the references therein. The special case τ n (t, s; η, θ) (1.2) considered 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 ∈ S 1 : η < argz < θ}.
By a simple computation, which will be recalled in Section 4, one obtains that the tau-functions (1.25) admit the expansion 29) are the so-called Plücker coordinates, and S i 1 ,...,i k (t) denote the Schur polynomials 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 τ n (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.
where A −− denotes the strictly lower triangular part of A, and D 1 and (D * 1 ) T (respectively D 2 and (D * 2 ) T ) represent the operator of derivation d/dz in the bases f n (z) n≥0 and h −1 n g n (z −1 ) n≥0 (respectively g n (z) n≥0 and h −1 n f n (z −1 ) n≥0 ), with f n (z), g n (z) the bi-orthogonal Laurent polynomials satisfying (1.18) and L(f m , g n ) = h n δ m,n .
2) On the tau-functions τ n (t, s), they are given by a centerless Virasoro algebra of partial differential operators in the (t, s) variables
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 biorthogonal polynomials {p (1) n (z), p (2) n (z)} n≥0 , given by the well known formulae with τ n = det µ kl 0≤k,l≤n−1 . Denoting by {f n , g n } n≥0 the sequence of monic right bi-orthogonal L-polynomials, multiplication by z in the bases (f n ) n≥0 and (g n ) n≥0 of C[z, z −1 ] will be represented by two pentadiagonal matrices A 1 and A 2 , which we call the generalized CMV matrices (and similarly of course for the sequence of left bi-orthogonal L-polynomials). Moreover, the entries of A 1 and A 2 will have simple expressions in terms of the variables x n and y n 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 L m,n := z m , z m+1 , . . . , z n−1 , z n , ∀m, n ∈ Z, m ≤ n, and for n ≥ 0 in the case of right bi-orthogonal L-polynomials. For left bi-orthogonal Lpolynomials the equivalent condition is We start by proving that sequences of right and left bi-orthogonal Lpolynomials for a given Toeplitz bi-moment functional L are closely related to each other. Proposition 2.3. Let f * n (z) = f n (z −1 ) and g * n (z) = g n (z −1 ). Then {f n , g n } 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 f n , g n ∈ L + n \ L + n−1 . Using the Toeplitz condition (1.11), the result then follows from 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.
2n+1 (z) = z n g 2n+1 (z).   An analogous statement holds for sequences {f n , g n } n≥0 of left bi-orthogonal L-polynomials, if we define Proof. For n ≥ 0, we define P n = 1, z, . . . , z n the vector subspace of polynomials with degree less than or equal to n, and P −1 := {0}. For {p (1) n , p Thus, according to Remark 2.2, {f n , g n } n≥0 is a sequence of right biorthogonal L-polynomials with respect to L if and only if {p 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 consequence 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 {f n , g n } n≥0 is a sequence of monic right bi-orthogonal L-polynomials with respect to L, i.e. the coefficients of z −n in f 2n , g 2n and z n+1 in f 2n+1 , g 2n+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).
The coefficients in the recurrence relations satisfy .

It follows from the definition of {g
Similarly we have This concludes the proof.
Corollary 2.7. With the same notations as in Theorem 2.6 we have Defining the vectors the five term recurrence relations obtained in Theorem 2.6 and Corollary 2.7 can be written in vector form where h = diag(h n ) n≥0 . We call the matrices A 1 , A 2 the (generalized) CMV matrices. Clearly, from (2.6), we have 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 variables x n , y n introduced in (1.15) entering the Szegö type recurrence relations (1.16).
Theorem 2.8. The non-zero entries of the CMV matrices A 1 and A 2 are By virtue of Theorem 2.4 we obtain As 2n−1 (z) is a monic polynomial of degree 2n+1, using the bi-orthogonality of the polynomials, we have 2n+1 (z) = 1.
is a polynomial of degree 2n, the first term is equal to 0 by bi-orthogonality. The remaining terms give By virtue of Theorem 2.4 we obtain 2n−1 (z) .
By using (1.16) and then (1.15) we have By virtue of Theorem 2.4 we obtain Using (1.16) we obtain The first term is equal to 0 as z 2n−2 p The other relations are proven in a similar way. This finishes the proof.

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) = (t 1 , t 2 , . . . , s 1 , s 2 , . . .) of the AL hierarchy by (t k ) k∈Z , with t −k = s k , k ≥ 1, and T 0 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 and let L be a quasi-definite bi-moment functional satisfying the Toeplitz condition. We introduce two matrices S 1 and S 2 by writing the vectors f (z), g(z) (2.4) of monic right bi-orthogonal L-polynomials with respect to L as follows With this definition, S 1 is a lower triangular matrix with all diagonal elements equal to 1, and S 2 is an upper triangular matrix such that h −1 S 2 has all diagonal elements equal to 1.
Associated to L we also define the semi-infinite bi-moment matrix with µ m,n as in (1.10), (1.12). The bi-moment matrix M can be written in terms of the vector χ(z) in (3.1) 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.
This can be written in matrix form

Using the expressions (3.2) we obtain
We define the semi-infinite shift matrix Λ by
Proof. We have The Ablowitz-Ladik hierarchy is defined on the space of bi-moments by the vector fields

The proof for
where we have put s k = t −k in (1.19). Obviously, these vector fields satisfy the commutation relations 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 For a square matrix A, we define • A 0 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 Proposition 3.1 in a product of a lower triangular and an upper triangular matrix.
Proof. On the one hand, we have using Proposition 3.
On the other hand, from equation (3.10) we have As Since ∂S 1 ∂t k 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 which establishes (3.11).
To establish the other formula, we write M = (S −1 Using the commutation relation (3.6) and (3.10), we also have , we obtain after some algebra Since (S −1 1 h) T is upper triangular, the first term in the right hand side of this equation is upper triangular. As S T 2 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 establishes (3.12), completing the proof.
We are now able to obtain a Lax pair representation for the Ablowitz-Ladik hierarchy.
By Lemma 3.3 we obtain 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 A 2 are obtained from those of A 1 by exchanging the roles of the variables x n and y n . Also A 1 contains as entries −x 2n+1 and y 2n and thus A 2 contains as entries x 2n and −y 2n+1 , n ≥ 0 (remember that x 0 = y 0 = 1). Thus the pair of Lax equations in (3.13) completely determines the Ablowitz-Ladik hierarchy in terms of the variables x n and y n .
Using the explicit expressions in terms of the variables x n and y n 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 T 1 and T −1 After the rescaling x n → e −2t x n , y n → e 2t y n , the vector field T 1 − 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 A 1 and A 2 . One can show that for k ≥ 1 k = 1 k T rA k 2 and T r denotes the formal trace, see [35] 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 V k , k ∈ Z, defined on the bi-moments by (1.21), on the CMV matrices (A 1 , A 2 ).
We first decompose the vector fields V k as follows where T k are the Ablowitz-Ladik vector fields (3.9). At the level of the bi-moments, the vector fields V k are given by These vector fields satisfy the following commutation relations It follows that Consequently, like the vector fields V k , the vector fields V k , 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 where and Λ is as in (3.5).
Remembering the notation (1.12), (3.15) writes d du k µ m,n = (m − n)µ m+k,n which is equivalent to the following equation on the bi-moment matrix M and, according to (2.6) and (2.7), these vectors satisfy

20)
We define the semi-infinite matrices D 1 , D * 1 and D 2 , D * 2 by the relations These matrices can be "dressed up" as explained in the next lemma.
Lemma 3.6. We have
Proof. By substituting the factorisation M = S −1 1 S 2 of the moment matrix into (3.18), we obtain Multiplying this equation on the left by S 1 and on the right by S −1 2 , we get (3.28) Using the factorisation of A 1 given in (3.7) and the factorisation of D 1 in (3.24), Term 1 gives

Similarly, the second term gives
where we have used the expression of D * 1 in Lemma 3.6. Substituting these results in (3.28), we obtain 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 which establishes (3.26).
To establish the other formula, we substitute the factorisation M = which follows from the commutation relation (3.6). This gives Multiplying this equation on the left by (S −1 1 h) −1 and on the right by (h −1 S 2 ) −1 , we get (3.29) Using the factorisation of A 2 in (3.7) and the factorisation of D 2 in (3.25), Term 2 gives Similarly, using the factorisation of A 2 in (3.7), gives Using the factorisation of A −1 2 in (3.8) and the factorisation of D * 2 in (3.25), we get Substituting these results in the transpose of (3.29), we obtain Since (S −1 1 h) T is upper triangular and S T 2 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 V k , 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 V k on the semi-infinite CMV matrices (A 1 , Using , which gives the equivalent formulation for the representation of the master symmetries on the CMV matrices (A 1 , A 2 ). 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 V k and V k in (3.14) and the Lax pair representation (3.30) of V k in Theorem 3.8, we have established the Lax pair representation (1.31), (1.32) of the vector fields V k as announced in Theorem 1.1 in the Introduction.
Using the explicit form of the CMV matrices (A 1 , A 2 ) 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 , V 0 , V 1 in terms of the variables x n , y n .

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 τ n (t, s) = det µ k−l (t, s) 0≤k,l<n .   This theorem is the key to the quick derivation of the various "Virasorotype" constraints satisfied by special tau-functions of the AL hierarchy. As an illustration we establish the following result.

Relabeling the indices as follows
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  Remark 4.3. After [24] was completed, we noticed 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 [29] 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 taufunction of the KdV hierarchy) are characterized by Virasoro constraints L k τ (t) = 0, k ≥ −1, corresponding to "half of" a Virasoro algebra, see [4,14,19,22,26,28,29] for the form of the operators L k 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 , a j |j ∈ Z} and { b , b j |j ∈ Z} and defining commutation relations with j, k ∈ Z. Let B be the space of formal power series in the variables t 1 , t 2 , . . . and s 1 , s 2 , . . . , and consider the following representation of A in B for j > 0, and µ ∈ C. Define the operators where k ∈ Z, a j , b j are as in (4.7) with µ = n, and where the colons indicate normal ordering, defined by and a similar definition for : b j b k :, obtained by changing the a's in b's in the former. Expanding the expressions in (4.8) we obtain for k > 0 and similar expressions for B (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 As shown in [25] (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 for k, l ∈ Z. Similarly, the operators B Proof. We give the proof for k, l ≥ 0, the other cases being similar. As j ] = 0, i, j ∈ Z, we have using (4.6), (4.10), (4.11) and (4.12) Relabeling the indices in the sums, we have 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 p i 0 ,...,i n−1 j 0 ,...,j n−1 . Next we shall compute the action of the Virasoro operators on the products S i n−1 −(n−1),...,i 0 (t)S j n−1 −(n−1),...,j 0 (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 x 1 , . . . , x n ∈ R n , we shall denote by |x 1 x 2 . . . x n | the determinant of the n×n matrix formed with the columns x i . Also, given two vectors x and y, x ∧ y denotes the usual wedge product, with components (x ∧ y) rs = x r y s − x s y r . Finally, for an n × n matrix A, A r will denote the rth column of A, and A T r the rth column of the transposed matrix, and tr(A) will mean the trace of A. With these conventions, we have the following lemma.
Proof. (i) Let A, B be n × n matrices, with A invertible. As A is invertible, its columns form a basis of C n and thus we have for a certain c (r) ∈ C n , whose components are c (ii) Using (4.14), we have We thus obtain where we have used the fact that for X, Y two n × n matrices, we have 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 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 (X T r ∧ X T s ) rs = (X r ∧ X s ) rs .
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.
Proof. We prove (i). Define the n × n matrices We have S i n−1 −(n−1),...,i 0 (t) = det A. It then follows that In the right-hand side, in the l th term, the l th and (l − 1) th columns coincide in the determinant, provided that l = 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.

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 S i n−1 −(n−1),...,i 0 (t)S j n−1 −(n−1),...,j 0 (s). We have the following lemma. Define the following n × n matrices and D = diag(n−1, n−2, . . . , 0). We shall denoteÂ(s) andB(j, s) the same matrices with t → s and (i 0 , . . . , i n−1 ) → (j 0 , . . . , j n−1 ). From the definition (1.26) of the elementary Schur polynomials it follows easily that for j ≥ 0, Consequently, by first using Leibniz's rule and then Lemma 4.5(i) we have for j ≥ 0 We are now ready to prove the lemma.
(i) From (4.9), we have L Combining both equations, we obtain (i).
(ii) From (4.9), we have L We compute, using (4.17) and (4.19) (iii) From (4.9), we have 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) We have and by a short computation

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. Proof. For simplicity, we will use the notations S i (t) = S i n−1 −(n−1),...,i 0 (t), S j (s) = S j n−1 −(n−1),...,j 0 (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 j n−l → j n−l − 1 gives 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 τ n (t), and Lemmas 4.9 and 4.10 we have for k = 0, 1, using the notations (4.27), 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)V k+1 τ n (s, t) =  Noticing that S j n−1 −(n−1),...,j 1 −1 (s) = 0 when j 1 = 0, and S j n−1 −(n−1),...,j 1 −1 (s) = S j n−1 −(n−1),...,j 1 −1,0 (s), when j 1 > 0, we get (4.36), and hence (4.37). This proves the case k = 2 and finishes the proof.
It would be nice to have a proof of Theorem 4.1 using the vertex operators techniques developed by the Kyoto school [13], but this remains a challenge for the future!