Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},\pi)$-type with the coefficients being the Pl\"ucker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For $\mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.


Introduction
Given a simple Lie algebra g over C, Drinfeld and Sokolov in [14] explained how to associate to it a family of commuting bi-hamiltonian PDEs known as the Drinfeld-Sokolov (DS) hierarchy of g-type. Nowadays, Drinfeld-Sokolov hierarchies are certainly among the most studied examples of integrable systems; one of their remarkable properties is that they are tau-symmetric [19,18,36,7], meaning that they admit the so-called tau function of an arbitrary solution to the hierarchy. For the case g = sl n (C) the DS hierarchy of g-type coincides (under a particular choice of the DS gauge [14,2]) with the Gelfand-Dickey hierarchy, and so, in particular, for n = 2, with the celebrated Korteweg-de Vries (KdV) hierarchy. It is known that tau functions of the Gelfand-Dickey hierarchies can be expressed as linear combinations of Schur polynomials with the coefficients being Plücker coordinates 1 [32,38,3,30]. In this short paper we aim to generalize this fact to an arbitrary given Lie algebra g. The generalization will depend on matrix realizations of g (note that the tau function itself is independent of the realizations of g [6]!). Indeed, one of our main observations is that the generalization of Schur polynomials are associated to faithful representations.
As an application of our result, we describe a systematic way of finding simple solutions (i.e. solutions whose tau function is a polynomial or a fractional power of it) of the DS hierarchy of g-type. Of course, in the case of the hierarchies of type A n , we recover the well-known results, since polynomial tau functions of these hierarchies (more generally of the KP hierarchy) had been studied for many years, due to their relations with Bäcklund transformations [1] and the dynamical systems of Calogero type (see for instance [35] and the references therein). Moreover, it had been proved that the polynomial tau functions of the so-called BKP hierarchy can be written in terms of the projective representations of the symmetric group [37] and this hierarchy, moreover, contains as reductions some of the DS hierarchies of D n -type, as explained in [12]. Nevertheless, it seems to us that a systematic approach to the study of polynomial tau functions associated to the general case (i.e. for an arbitrary Lie algebra) is still missing, and this paper gives a first result in this direction. The polynomial tau functions we obtain are, actually, quite non-trivial, and can also be used to give some explicit information about the structure of the bilinear equations for the hierarchy.
In order to state precisely our results, we need to fix some notations about finite dimensional Lie algebras, loop algebras and Toeplitz determinants. Let g be a simple Lie algebra over C of rank n, and h, h ∨ the Coxeter and dual Coxeter numbers, respectively. Fix h a Cartan subalgebra of g. Take Π = {α 1 , . . . , α n } ⊂ h * a set of simple roots, and let △ ⊂ h * be the root system. We know that g has the root space decomposition Let θ denote the highest root with respect to Π, and (·|·) : g × g → C the normalized Cartan-Killing form, i.e. (θ|θ) = 2. For a root α ∈ △, denote by H α the unique vector in h satisfying (H α |H β ) = (α|β), ∀ β ∈ △.
be a set of Weyl generators of g. They satisfy where (A ij ) n i,j=1 is the Cartan matrix of g. Choose E −θ ∈ g −θ , E θ ∈ g θ , normalized by the conditions (E θ | E −θ ) = 1 and ω(E −θ ) = −E θ , where ω : g → g is the Chevalley involution. Let I + := n i=1 E i be a principal nilpotent element of g. Denote by L(g) = g ⊗ C[λ, λ −1 ] the loop algebra of g. On L(g) there is the principal gradation defined by assigning Here, elements in L(g) j have degree j. Define Λ ∈ L(g) by Clearly, Λ is homogeneous of degree 1. Denote by L(g) <0 elements in L(g) with negative degrees, similarly, by L(g) ≤0 elements with non-positive degrees.
It was shown in [26,29] that Ker ad Λ ⊂ L(g) has the following decomposition where the integers m 1 , . . . , m n are the exponents of g, and E is called the set of exponents of L(g). We use E + to denote the set of positive exponents. The elements Λ i commute pairwise They can be normalized by In particular, we can choose Λ 1 = Λ.
Let us now take π : g → gl(m, C) an irreducible faithful representation. When no confusion can arise, for b ∈ g, we write π(b) simply as b. Our generalization will be based on the infinite Grassmannian approach [32,33] and the related Plucker coordinates.
Definition 1.1. Let ξ := ℓ∈E + t ℓ Λ ℓ with t ℓ , ℓ ∈ E + being indeterminates and let s denote the Laurent matrix associated with e ξ , namely, The Schur polynomials of (g, π)-type are labelled by partitions and defined by Definition 1.2. In the case π is taken as the adjoint representation of g, we call s λ , λ ∈ Y the intrinsic Schur polynomials of g-type.
, denote by r X the Laurent matrix associated with e X , i.e. r X := L(e X ).
Remark 1.6. The matrix-valued affine coordinates Z X, IJ and their generating formula (9) were introduced in [3] by F. Balogh and one of the authors of the present paper for the sl 2 (C) case.
The following theorem is the main result of the paper. Denote by κ the constant such that (a|b) = κ Tr(π(a)π(b)) ∀a, b ∈ g.
Theorem 1.7. For any X ∈ λ −1 g[[λ −1 ]], the formal series τ defined by is a tau function of the Drinfeld-Sokolov hierarchy of g-type. Moreover, s ν and r X,ν have the following expressions We refer to (11)-(13) as the Sato-Zhou type formula for tau functions of the DS hierarchy. Remark 1.8. As the reader might already have noticed, here the terminology is very similar to the one used to deal with the KP hierarchy in the Sato's approach. However, it is worth mentioning that tau functions of the DS hierarchies of g-type in general are not KP tau functions (except for g = sl n+1 (C)). One way to see it (which is close to the spirit of this paper) is that the generalized Schur polynomials s ν of (g, π)-type we defined are "reductions" (in the sense of the Remark 1.3) of the usual ones [30] just in the A n case.
Remark 1.9. The formula (11) is intrinsic when π is taken as the adjoint representation of g.
We will study the intrinsic Schur polynomials associated to g in a future publication.
Remark 1.10. For the ABCD cases, a result similar to Theorem 1.7 was obtained in [39] where a different method was used; see also in [4] for more details for the A n case.
Organization of the paper In Section 2 we review the Drinfeld-Sokolov hierarchies and their tau functions. In Section 3 we prove Theorem 1.7. Some explicit examples and applications are given in Section 4. A list of first few Schur polynomials of (g, π)-type for g of low ranks and particular choices of π are given in the Appendix.

Review of the Grassmannian approach to the DS hierarchy
Denote by b the Borel subalgebra of g, i.e. b := g ≤0 , and by n the nilpotent subalgebra n := g <0 . Define a linear operator L by where q(x) ∈ b. It is proved by V. G. Drinfeld and V. V. Sokolov [14] that there exists a unique smooth function U(x) ∈ g((λ −1 )) <0 ∩ Im ad Λ such that The following commuting system of PDEs are called the pre-DS hierarchy of g-type.
Gauge transformations. For any smooth function N(x) ∈ n, the map Below we fix V a DS gauge. It was observed in [14] that the flows (15) can be reduced to gauge equivalent classes; moreover, for any q( Take v 1 , . . . , v n a homogeneous basis of V , namely deg v i = −m i , and write The DS hierarchy of g-type is defined as the system of the pre-DS flows for the complete set of representatives (aka gauge invariants) u 1 , . . . , u n . Clearly, the precise form 2 of this integrable hierarchy depends on the choice of the DS gauge V ; the hierarchies under different choices of V are Miura equivalent [?, 24,25,19,6]. We remark that a unified algorithm of writing the DS hierarchy of g-type for an arbitrary choice V was obtained recently in [6]; it has the form where a i,ℓ [u 1 , . . . , u n ] are differential polynomials of u 1 , . . . , u n . It should also be noted that for the DS hierarchy of g-type the time variable t 1 can be identified with −x.
The hierarchy (17) is known to be Hamiltonian and tau-symmetric [19,24,36,7]. Therefore, for an arbitrary solution q can of (17), there exists a tau function τ (t) of q can . The tau function is determined up to a multiplicative factor of the form where c ℓ are arbitrary constants. We review in this subsection the Grassmannian approach to tau functions.
Here A k,ℓ,i are called the affine coordinates [20] of W .
g as the following subset of the big cell Gr (0) We call Gr (0) g the embedded big cell of g-type. For The matrices G 0 , G 1 , . . . serve as the matrix-valued coordinates for the point W corresponding to a; see Fig. 1. Clearly, G 0 = I.
The following theorem comes from the results obtained in [9,10].
where κ is defined in (10). Then τ is a tau function of the DS hierarchy associated to g.
The stabilization proved in [22] for the case of the Witten-Kontsevich tau function and extended in [10] for the general cases ensures that the limit in (18) is meaningful.
3 Proof of Theorem 1.7 Define γ = e ξ e X , where we recall that X is the given element in We have L(γ) = L(e ξ ) L(e X ) = s r X where s, r X are defined in (5), (7), respectively. For any N ≥ 1, define two matrices By using the well-known Cauchy-Binet formula (see for instance [21]) we obtain [32,20] from where we recall that r X,λ and s λ are defined by As explained in [3], formulae (8) and (9) give the Gaussian eliminations and formulae (12) and (13) are due to the Giambelli-type formula [20,30,3]. The theorem is proved.

Polynomial tau functions and bilinear equations
Theorem 1.7 gives a simple procedure to compute the tau function τ when τ 1/κ is a polynomial. Indeed, let us fix the Lie algebra g and take a faithful representation π. Choosing X ∈ λ −1 g[[λ −1 ]] such that π(X) is a nilpotent matrix, the infinite series in (11) becomes finite, as it is easy to verify that only finitely many Plücker coordinates {r µ , µ ∈ Y} are non zero. Consequently, τ 1/κ is polynomial. This simple idea was used for example in [3] for the KdV hierarchy. If κ = 1, then the tau function itself is a polynomial. Interestingly enough, in the computations we will perform, even when κ = 1/2, we obtain some polynomial tau functions: in other words, the finite sum in (11) is a perfect square. Even if this result has not been proved in general, we expect that our procedure gives a systematic way to compute all the polynomials tau functions (up to a shift of the time variables {t i , i ∈ E + }) of the DS hierarchy of g-type. As stated in the introduction of [28], this is an interesting open problem.
In what follows we compute the first few polynomial tau functions of the DS hierarchy of g-type for g = A 1 , A 2 , B 2 and D 4 . We use these particular tau functions to deduce possible bilinear equations of small degrees. Note that each Drinfeld-Sokolov hierarchy has infinitely many solutions. The usual question is to find particular solutions to the DS hierarchy (solve all PDEs in this hierarchy together). Here we consider the inverse: Deduce possible PDEs from particular solutions. Sometimes, one particular solution already contains all the information of an equation and of the whole hierarchy. For example, the "topological solution" was used by B. Dubrovin and Y. Zhang to construct the integrable hierarchy of topological type [19,17]. However, a polynomial tau function τ poly of the DS hierarchy contains less information, namely, if τ poly satisfies some PDE, it will not guarantee directly that other tau functions of the DS hierarchy satisfy this PDE. Nevertheless, if τ poly does not satisfy a PDE, then the PDE cannot belong to the DS hierarchy.

Bilinear derivatives
Given two smooth functions f (x), g(x) with independent variables x = (x i ) i∈I , where I denotes an index set. The bilinear derivatives D i 1 · · · D i k are operators defined via the identity It means that, expanding both sides of this identity in h and comparing the coefficients of monomials of h, we obtain, for example, For the Drinfeld-Sokolov hierarchy of g-type, we take I := E + . There is a natural gradation for the bilinear derivatives, defined by assigning deg D i = i for i ∈ E + . Denote by H g the linear space of bilinear equations satisfied by the Drinfeld-Sokolov hierarchies of g-type, which decomposes into homogeneous subspaces The gradation allows us to list all possible bilinear equations up to certain degree.

Examples of polynomial tau functions 4.2.1 The A 1 case
Let us chose the standard matrix realization g = sl(2; C). Consider the following two elements The associated polynomial tau functions are respectively. Similarly, one computes polynomial tau functions corresponding to elements of the form λ −k F , λ −k E, k ≥ 2. For example, for k = 2, we obtain corresponding to λ −2 F and λ −2 E, respectively. Now consider all bilinear equations up to degree 4: where β, α 0 , α 1 , α 2 are complex constants. Requiring that τ 1 , τ 2 satisfy the above ansatz (23), we find that up to a multiplicative constant there is only one possible choice of coefficients: Similarly up to degree 6, we find out only two more possible linearly independent bilinear equations that are satisfied by τ 1 , τ 2 , τ 3 , τ 4 which are well known to belong to the hierarchy of A 1 -type, that is the KdV hierarchy. Consequently, we have shown that dim C H Relation with the Adler-Moser polynomials. An alternative way of computing polynomial tau functions for the KdV hierarchy was given by Adler and Moser [1]. Define a family of polynomials θ k (x = q 1 , q 3 , q 5 , . . . , q 2k−1 ), k ≥ 0 recursively by where the prime denotes the x-derivative and for each k ≥ 2 the integration constant is chosen to be q 2k−1 . The polynomials θ k are known as the Adler-Moser polynomials. It was also proven in [1] that there exists a unique change of variables q → t that transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarychy. In [15], one of the authors of the present paper proved that the desired change of variables is given by where α 2i−1 := (−1) i−1 3 2 · · · (2i − 3) 2 (2i − 1). Up to a shift and renormalisation of the times, we recover in particular the polynomials given in equations (20)-(22).

The A 2 case
We still chose the standard matrix realization g = sl(3; C). Consider for example the following two elements in λ −1 g[[λ −1 ]] : where a 1 , a 2 , a 3 are arbitrary constants. The corresponding polynomial tau functions will be denoted by τ 1 , τ 2 , respectively. We have    Consider all possible bilinear equations of degree 4: Requiring τ 1 satisfies this ansatz we find that there is only one possible choice: Similarly, requiring that τ 1 and τ 2 to both satisfy the ansatz of bilinear equation of degree 6, we find that there are only two linearly independent bilinear equations of degree 6: which are well known to belong to the hierarchy of A 2 -type (i.e. the Boussinesq hierarchy).

The B 2 case
We chose the matrix realization of the B 2 simple Lie algebra as in [14]. We consider two explicit examples given respectively by the following matrices 3 The associated tau functions will be denoted by τ 1 and τ 2 . They have the expressions  Consider all bilinear equations up to degree 4 where α 0 , . . . , α 3 are constants. Requiring that τ 1 satisfies this ansatz of bilinear equations we find that there is no solution. Similarly, up to degree 8, we find that there are only two possible homogeneous equations (one is of degree 6 and the other is of degree 8). We arrive at Our last remark is that under the following linear change of time variables the bilinear equations (29)-(31) in the new time variables T 1 , T 3 , T 3 ′ , T 5 coincide with those of Kac and Wakimoto [28]. Essentially speaking such a change of times is simply a renormalization of flows.
A List of generalized Schur polynomials of (g, π)-type Take π as in [14,10]. We list in Table 1 the first several Schur polynomials of (g, π)-type for simple Lie algebras of low ranks.   Table 1: Simple Lie algebras and Schur polynomials of (g, π)-type