Johan van de Leur

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.

This, to a large extent, expository paper describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X = A, B, C, or D, and AKP = KP and their reductions, associated with the conjugacy classes of the Weyl groups of classical Lie algebras of type X. As usual, the main tool is the multicomponent boson–fermion correspondence, which leads to the corresponding tau-functions, wave functions, dressing operators, and Lax operators.

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the Kadomtsev-Petviashvili (KP) hierarchy: by a generalized Jacobi-Trudi formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its n-reductions. In particular, for n = 3 we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the KP hierarchy: by a generalized Jacobi-Trudy formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its n-reductions. In particular, for n=3 we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.

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We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin--Frobenius manifold is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.

It is well-known that a B type Kadomtsev–Petviashvili (BKP) tau-function is the square root of a certain Kadomtsev–Petviashvili (KP) tau-function, provided one puts the even KP times equal to zero. In this paper we compute for all polynomial BKP tau-function its corresponding KP ‘square’. We also give, in the polynomial case, a representation theoretical proof of a recent result by Alexandrov, viz that a Korteweg–de Vries (KdV) tau-function becomes a BKP tau-function when one divides all KdV times by 2.

We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero-mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial tau-functions of these hierarchies in terms of Schur polynomials and Q-Schur polynomials, respectively. We also obtain formulas for polynomial tau-functions of the reductions of the s-component KP hierarchy associated with partitions in s parts.