For the KP hierarchy, a formal power series is a tau function if and only if the coefficients of its Schur-function expansion satisfy a Giambelli-type determinantal identity, and the analogous statement holds for the BKP hierarchy with Schur Q-functions; both characterizations were proved by Shigyo directly from the addition formulae, without Sato theory. Gao, Li and He (2016) then established addition formulae equivalent to the discrete KP, q-deformed KP, two-component BKP, and D-type Drinfeld-Sokolov hierarchies and announced the Giambelli-type formulae for exactly these hierarchies as their next objective. This announced extension has not been carried out in the literature for any of these four hierarchies, the known Giambelli-type results covering only the KP, CKP, and single-component BKP cases. The open problem is, for each of these four hierarchies, to identify the Schur-type basis of the tau-function expansion and to prove that a series is a tau function precisely when its expansion coefficients satisfy the corresponding determinantal or Pfaffian Giambelli-type identity, the D-type Drinfeld-Sokolov case following from the two-component BKP case by reduction.

Background

The Kadomtsev-Petviashvili (KP) hierarchy is the prototypical soliton hierarchy: an infinite family of commuting nonlinear evolution equations encoded in a pseudo-differential Lax operator (a differential operator extended to allow negative powers of the derivative ∂x\partial_x), all of whose solutions are captured by a single tau function τ(t1,t2,… )\tau(t_1,t_2,\dots) satisfying an infinite system of Hirota bilinear equations, that is, quadratic identities in which each differential operator acts symmetrically on two copies of τ\tau. An addition formula is a finite identity expressing tau functions whose arguments are shifted by two families of spectral parameters (k+1k+1 parameters αi\alpha_i and k−1k-1 parameters βj\beta_j) as an alternating (k+1)(k+1)-term sum of products of tau functions with smaller shifts; in the simplest case it degenerates to the classical Fay identity of the KP hierarchy. The deep structural fact, established for KP, for the modified KP (mKP) and BKP hierarchies by Shigyo, and for four further hierarchies by Gao, Li and He, is that the addition formula is equivalent to the full hierarchy: a function is a tau function if and only if it satisfies the addition formula. Here BKP denotes the KP hierarchy of type B, whose tau-function theory is Pfaffian rather than determinantal (a Pfaffian is the signed square root of the determinant of a skew-symmetric matrix, the analogue of a determinant for odd-sized constructions).

A second thread connects tau functions to symmetric functions. Any formal power series in the time variables expands in Schur functions sλs_\lambda, the symmetric functions indexed by partitions that arise as characters of the symmetric groups; the classical Giambelli formula writes sλs_\lambda as a determinant of hook Schur functions. Shigyo proved, using addition formulae rather than the Sato-Segal-Wilson infinite-Grassmannian picture, that a formal power series with constant term 1 is a KP tau function precisely when its Schur-expansion coefficients satisfy a Giambelli-type determinantal formula, and established the analogous statement for the BKP hierarchy in terms of Schur Q-functions QλQ_\lambda (the symmetric functions governing projective representations of the symmetric groups, which satisfy Pfaffian identities; determinantal and Pfaffian forms for composite Schur and Q-functions go back to the boson-fermion treatment of Jarvis and Yung). Shigyo later extended the BKP result to series with vanishing constant term.

Gao, Li and He proved addition formulae — and their equivalence with the corresponding hierarchies of Hirota bilinear equations — for four hierarchies to which this Giambelli-type theory had not been extended: (i) the discrete KP hierarchy, in which a difference operator replaces the pseudo-differential operator and the tau function τ(n;t)\tau(n;t) depends on a lattice index nn as well as on continuous times; (ii) the q-deformed KP hierarchy, built on q-calculus (the q-derivative Dqf(x)=f(qx)−f(x)(q−1)xD_q f(x)=\frac{f(qx)-f(x)}{(q-1)x} replacing ddx\frac{d}{dx}) with tau function τq(x;t)\tau_q(x;t); (iii) the two-component BKP hierarchy, the B-type analogue of the two-component KP hierarchy built from two sets of odd time variables, associated with the infinite-dimensional orthogonal Lie algebra D∞D_\infty (the type-D counterpart of the algebra b∞b_\infty underlying the single-component BKP hierarchy) and connected to the Novikov-Veselov equation (a two-dimensional analogue of the Korteweg-de Vries equation) and to the total descendant potential of D-type singularities; and (iv) the D-type Drinfeld-Sokolov hierarchy (a member of the family of integrable hierarchies constructed from affine Lie algebras of type D), which they obtain as a reduction of the two-component BKP hierarchy, so that its tau function τds(t,tˉ)\tau_{\mathrm{ds}}(t,\bar t) and addition formula descend from the two-component BKP ones. Their paper closes by announcing, as the next objective, the study of the Giambelli type formula in exactly these hierarchies. That announced extension — carried out in the literature for KP, for BKP, and (for polynomial tau functions) for the CKP hierarchy (the type-C analogue of the BKP hierarchy), but for none of these four — is the open problem stated below.

Problem Statement

For each of the four hierarchies for which Gao, Li and He proved addition formulae — the discrete KP, the q-deformed KP, the two-component BKP, and (obtained by reduction from the two-component BKP) the D-type Drinfeld-Sokolov hierarchy — establish the Giambelli-type formulae for tau functions, mirroring the known single-component theory. Concretely: (i) identify the Schur-type symmetric-function basis in which the tau function of the hierarchy admits a series expansion (for the two-component BKP hierarchy a two-variable analogue of the Schur Q-functions is the natural candidate; for the discrete and q-deformed hierarchies the appropriate basis and expansion framework, accommodating the lattice index nn of τ(n;t)\tau(n;t) and the q-shift structure of τq(x;t)\tau_q(x;t) respectively, must be determined); (ii) derive the corresponding Giambelli-type determinantal or Pfaffian identities satisfied by the expansion coefficients; and (iii) prove the characterization theorem: a formal power series with nonzero constant term is a tau function of the hierarchy if and only if its expansion coefficients satisfy the Giambelli-type identity. The single overarching objective is to extend the addition-formula-based coefficient characterization already established for the KP and BKP hierarchies to these four hierarchies, for which the addition formulae of Gao, Li and He are the natural starting point; the four cases form the coherent uniform family defined by that paper, the D-type Drinfeld-Sokolov case is obtained by reduction of the two-component BKP result rather than being an independent deliverable, and the three numbered components per hierarchy jointly support one characterization result.

References

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