Polynomial tau-functions of the n -th Sawada-Kotera hierarchy

We ﬁnd all polynomial tau-functions of the n -th reduced BKP hierarchy (= n -th Sawada-Kotera hierarchy). The name comes from the fact that for n = 3 the simplest equation of the hierarchy is the famous Sawada-Kotera equation.

For even k this equation is trivial, L(t, ∂) is given by (1), and equation (6) for k = 3 is the KdV equation [2].
Recall that, in order to construct solutions of the KP hierarchy and the reduced KP hierarchies one introduces the tau-function τ (t), defined by [13], [2]: where P (t, ∂) is a pseudodifferential operator, with the symbol The tau-function has a geometric meaning as a point on an infinite-dimensional Grassmannian, and in [13] Sato showed that all Schur polynomials are tau-functions of the KP hierarchy.Recently, all polynomial tau-functions of the KP hierarchy and its n-reductions have been constructed in [8] (see also [11]).
The CKP hierarchy (KP hierarchy of type C) can be constructed by making use of the KP hierarchy, and assuming the additional constraint L( t, ∂) * = −L( t, ∂) (see e.g.[10] for details).Its 3-reduction is defined by the constraint L( t, ∂) = L( t, ∂) 3 is a differential operator, and the corresponding hierarchy is where L is given by (2).The simplest non-trivial of these equations arizes for k = 5, and it is the Kaup-Kupershmidt equation.All polynomial tau-functions of (9) (and all n reductions of the CKP hierarchy) have been constructed in [10].
In the present paper we construct all polynomial tau-functions of the n-reduced BKP hierarchies (KP hierarchy of type B).These are hierarchies of Lax equations on the differential operator satisfying the constraint The n-th reduced BKP hierarchy is We call it the n-th Sawada-Kotera hierarchy, since for n = 3, L is given by (3), and for k = 5, equation ( 12) is the Sawada-Kotera equation [14] (see equation ( 33)).
In the present paper, using the description of polynomial tau-functions of the BKP hierarchy [9], [11] (see Theorem 2), we find all polynomial tau-functions for the n-th Sawada-Kotera hierarchies (see Theorem 9), and, in particular, for the Sawada-Kotera hierarchy (see Corollary 11).

The BKP hierarchy and its polynomial tau-functions
In this section we recall the construction of the BKP hierarchy [3] and description of its polynomial tau-functions from [9] and [11].
Following Date, Jimbo, Kashiwara and Miwa [3] (see also [9] for details) we introduce the BKP hierarchy in terms of the so called twisted neutral fermions φ i , i ∈ Z, which are generators of a Clifford algebra over C, satisfying the following anti-commutation relatons Define a right (resp.left) irreducible module F = F r (resp.F l ) over this algebra by the following action on the vacuum vector |0 (resp.0|) The quadratic elements form a basis of the infinite-dimensional Lie algebra so ∞,odd over C. Let SO ∞,odd be the corresponding Lie group.We proved in [7,Theorem 1.2a] that a non-zero element τ ∈ F lies in this Lie group orbit of the vacuum vector |0 if and only if it satisfies the BKP hierarchy in the fermionic picture, i.e., the following equation in Non-zero elements of F , satisfying (15), are called tau-functions of the BKP hierarchy in the fermionic picture,.
The group SO ∞,odd consists of elements G leaving the symmetric bilinear form Stated differently, The group orbit of the vacuum vector is the disjoint union of Schubert cells (see Section 3 of [9] for details).These cells are parametrized by the strict partitions Namely, the cell, attached to the partition λ is An element τ ∈ C λ corresponds to the following point in the maximal isotropic Grassmannian (i.e. a maximal isotropic subspace of V = j∈Z Cφ j ): For instance Ann |0 = span{φ 1 , φ 2 , . ..}.
Using the bosonization of the equation ( 15), one obtains a hierarchy of differential equations on τ ([3], [4], [11] or [9,Section 3]).This bosonization is an isomorphism σ between the spin module F and the polynomial algebra Explicitly, introduce the twisted neutral fermionic field and the bosonic field where the normal ordering : : is defined by The operators α j satisfy the commutation relations of the Heisenberg Lie algebra and its representation on F is irreducible [7,Theorem 3.2].Using this, we obtain a vector space isomorphism σ : F → B, uniquely defined by the following relations: Since ( 15) can be rewritten as under the isomorphism σ the equation ( 15) turns into: where t = (t 1 , t 3 , t 5 , . ..) and t′ = (t ′ 1 , t ′ 3 , t ′ 5 , . ..).Therefore, τ ( t) is the vacuum expectation value Furthermore, by making a change of variables, as in [4, page 972], viz.
, and using the elementary Schur polynomials s j (r), which are defined by exp we can rewrite (24), where we assume where ỹ = (y 1 , 0, y 3 , 0, . ..) and ∂y = ( ∂ ∂y 1 , 0, Using the notation p(D)f The simplest equation in this hierarchy is [4, Appendix 3]: If we assume that our tau-function does not depend on t 3 , then this gives Letting x = t 1 , t = 1 9 t 5 , and and viewing the remaining t j as parameters, equation (31) turns into the famous Sawada-Kotera equation [14]: Another approach is by using the wave function, see [3, page 345], where P ( t, z) = 1 + ∞ j=1 p j ( t)z −j , and in particular Letting P ( t, ∂) be the pseudodifferential operator in ∂ = ∂ ∂t 1 with the symbol P ( t, z), equation (24) turns into Now, using the fundamental lemma, Lemma 1.1 of [2] or Lemma 4.1 of [6], we deduce from (36): Next, introducing the Lax operator we deduce from (37) that L satisfies [3] Note that, since u 1 ( t) = − ∂p 1 ( t) ∂t 1 and the fact that p 1 (t) is given by (35), we find that which explains the choice (32) of u(x, t) to obtain the Sawada-Kotera equation from the Hirota bilinear equation (31).
To obtain the second equation of (38), we use (37) and the first equation of (38), which is equivalent (see [3], page 356) to the fact that L( t, ∂) 2j−1 , for j = 1, 2, . .., has zero constant term.Let us prove that the first equation of (38) indeed implies this fact.We have Now, using the fact that the constant term of L k is equal to we find that the constant term of L k is zero whenever k is odd.
Remark 1 Note that this also means that we can replace the second equation of (38) by, cf.[12], In the formulation of Kupershmidt [12], this means that L satisfies not only the KP equation for the odd times, but also his formulation of the modified KP hierachy (only for the odd times).
Remark 3 The connection between the set of strict partitions and the extended strict partitions is as follows.If λ = (λ 1 , λ 2 , . . ., λ k ) is a strict partition and k is even, then this partition is equal to the extended strict partition λ.However, if k is odd, the Pfaffian of a k × k anti-symmetric matrix is equal to 0, hence in that case we extend λ by the element 0, i.e., the corresponding extended strict partition is then (λ 1 , λ 2 , . . ., λ k , 0).

The n-th Sawada-Kotera hierarchy and its polynomial tau-functions
As we have seen in Section 2, a necessary condition for a tau-function to give a solution of the Sawada-Kotera equation is that ∂τ ( t) ∂t 3 = 0.This means that the tau-function lies in a smaller group orbit of the vacuum vector |0 .Instead of the SO ∞,odd orbit of the vacuum vector |0 , we consider the twisted loop group G  3 , to obtain the 3-reduced BKP hierarchy [3].More generally (see also [3]), when n = 2k + 1 > 1 is odd, the 2k + 1-reduced hierarchy is related to the twisted loop group G 2k [1], [7] (see [5,Chapter 7] for the construction of these Lie algebras).Elements G in this twisted loop group not only satisfy (18), which implies j∈Z (−1) j a kj a ℓ,−j = (−1) k δ k,−ℓ , but also the n-periodicity condition a i+n,j+n = a ij .This means that these group elements also commute with the operator Since i∈Z (−1) pn−i φ i |0 ⊗ φ pn−i |0 = 0, we find that the elements τ in the orbit of the vacuum vector of this twisted loop not only satisfy (15), but also satisfy the conditions j∈Z (−1) pn−j φ j τ ⊗ φ pn−j τ = 0, p = 1, 2, . . . .
From this we find that the tau-function satisfies Since we consider only polynomial tau-functions, we find that for odd n: If n is even there is no such restriction, because the Sato-Wilson equation ( 37) is only defined for odd flows.However, the additional equations (43) still hold and give additional constraints on the tau-function.
Remark 5 If n is odd.Proposition 4 gives that a polynomial BKP tau-function is n-th Sawada-Kotera tau-function if and only if τ satisfies ∂τ ∂tn = 0. Since L satisfies the BKP hierarchy L = L n also satisfies the BKP hierarchy.For n = 3, assuming the constraint that L is a differential operator, L is given by (3) and 3 ) ≥0 , L] is the Sawada-Kotera equation (33).This leads to the following definition.11).The system of Lax equations is called the n-th reduced BKP hierarchy or the n-th Sawada-Kotera hierarchy.For n = 3 it is called the Sawada-Kotera hierarchy.
The geometric meaning of equation ( 42) is that the space Ann τ is invariant under the shift Λ n , where Λ n (φ i ) = φ i+n .
As in the SO ∞,odd case, all polynomial tau-functions in this n-reduced case lie in some Schubert cell.Such a Schubert cell has a "lowest" element w λ , for λ a certain strict partition.This element can be obtained from the vacuum vector by the action of the Weyl group corresponding to G (2) n .The element lies in the Weyl group orbit of |0 , corresponding to SO ∞,odd , see [15], however not all such elements lie in the Weyl group orbit of |0 for G n .For this, consider Ann The element w λ lies in the G n Weyl group orbit if and only if Ann w λ is invariant under the action of Λ n , which means that the (λ 1 + 1 shifted) set V λ = {λ 1 + λ i + 1|, i = 1, . . ., k} ∪ {λ 1 − j + 1| 0 < j < λ 1 , j = λ i for i = 1, . . ., k}, (51) must satisfy the −n shift condition, i.e., if µ j ∈ V λ , then µ j − n ∈ V λ or µ j − n ≤ 0. (52) Only the elements w λ , for which the corresponding V λ satisfies condition (52) lie in the G n group orbit.Remark 8 Note that (52) means that λ is a strict partition which is the union of strict partitions (nm + a i , n(m − 1) + a i , , . . ., n + a i , a i ), with 1 ≤ a i < n and 1 ≤ i < n, such that a j − n = −a ℓ .In other words a j + a ℓ = n.Hence there are at most n 2 − 1 such a i .
To a strict partition λ, that satisfies condition (52), the corresponding Schubert cell is then obtained through the action on a w λ by an upper-triangular matrix in the group G (2) n .This produces, up to a constant factor, elements (v j , v ℓ ) = 0, for j, ℓ = 1, . . ., k, and if λ i = λ j − n, then v i = Λ n (v j ). (54) We first express the constants a ij in terms of other constants by letting a ij = s i+λ j (c λ j ), where the s i are elementary Schur polynomials.
to the affine Lie algebra sl