The Schur function expansion of Sato-Segal-Wilson KP tau-functions is reviewed. The case of tau-functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Plücker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann theta function or Klein sigma function along the KP flow directions. Using the fundamental bi-differential, it is shown how the coefficients can be expressed as polynomials in terms of Klein's higher genus generalizations of Weierstrass' zeta and P functions. The cases of genus two hyperelliptic and genus three trigonal curves are detailed as illustrations of the approach developed here.
Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau-function as a linear expansion of Schur Q-polynomials. In this paper, we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints, and consequently obtain a proof of this formula without using matrix model. We also give a proof for Alexandrov's conjecture that Kontsevich-Witten tau-function is a hypergeometric tau-function of the BKP hierarchy after re-scaling.
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 to partitions in s parts.
A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear partial differential equations. Specifically, just as the Schur polynomials are used to expand tau-functions as a sum, it is shown that it is natural to expand a quotient of tau-functions in terms of these generalized Schur functions. The coefficients in this expansion are found to be constrained by the Plücker relations of a grassmannian.
We introduce hypergeometric functions related to projective Schur functions $Q_λ$ and describe their properties. Linear equations, integral representations and Pfaffian representations are obtained. These hypergeometric functions are vacuum expectations of free fermion fields, and thus these functions are tau functions of the so-called BKP hierarchy of integrable equations.
In this paper, we prove a conjecture of Alexandrov that the generalized Brezin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients.
The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.
We study Slavnov's inner products for open Temperley-Lieb chains and their relations with the KP hierarchy. We show that when (s=1/2) the quantum group invariant XXZ spin chain has Slavnov products given by the quotient of tau functions. The Schur polynomials expansion and a matrix construction for the Baker-Akhiezer functions are also briefly considered.
We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. The role of additional symmetries in generating hypergeometric tau-functions is explained.
We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the integral over eigenvalues, which in turn is certain tau function. The resulting tau function may be analyzed either by the method of orthogonal polynomials or by the Schur functions expansion method.
We give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a $q$-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and $q$-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguer…
In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial $s_λ(t)$ by certain shifts of arguments. In the present paper we give a simpler proof of this result, using the (1-component) boson-fermion correspondence. Moreover, we show that this approach can be applied to the s-component KP hierarchy, using the s-component boson-fermion correspondence, finding thereby all its polynomial tau-functions. We also find all polynomial tau-functions for the reduction of the s-component …
In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar conjecture for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.
We study bosonic tau functions in relation with the charged free bosonic fields. It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector, and some tau functions were given in the completion of M using Schur functions. We also give a new proof of Borchardt's identity and obtain several q-series identities by using the boson-boson correspondence.
We consider multiple sums and multi-integrals as tau functions of the BKP hierarchy using neutral fermions as the simplest tool for deriving these. The sums are over projective Schur functions $Q_α$ for strict partitions $α$. We consider two types of such sums: weighted sums of $Q_α$ over strict partitions $α$ and sums over products $Q_αQ_γ$. In this way we obtain discrete analogues of the beta-ensembles ($β=1,2,4$). Continuous versions are represented as multiple integrals. Such sums and integrals are of interest in a number of problems in mathematics and physics.
We consider Schur function expansion for the partition function of the model of normal matrices. We show that this expansion coincides with Takasaki expansion \cite{Tinit} for tau functions of Toda lattice hierarchy. We show that the partition function of the model of normal matrices is, at the same time, a partition function of certain discrete models, which can be solved by the method of orthogonal polynomials. We obtain discrete versions of various known matrix models: models of non-negative matrices, unitary matrices, normal matrices.
We introduce a useful and rather simple class of BKP tau functions which which we shall call "easy tau functions". We consider two versions of BKP hierarchy, one we will call "small BKP hierarchy" (sBKP) related to $O(\infty)$ introduced in Date et al and "large BKP hierarchy" (lBKP) related to $O(2\infty +1)$ introduced in Kac and van de Leur (which is closely related to the large $O(2\infty)$ DKP hierarchy (lDKP) introduced in Jimbo and Miwa). Actually "easy tau functions" of the sBKP hierarchy were already considered in Harnad et al, here we are more interested in the lBKP case and also the…
We give simple linear algebraic proofs of Eynard-Mehta theorem, Okounkov-Reshetikhin formula for the correlation kernel of the Schur process, and Pfaffian analogs of these results. We also discuss certain general properties of the spaces of all determinantal and Pfaffian processes on a given finite set.
We investigate the Brezin-Gross-Witten model, a tau-function of the KdV hierarchy, and its natural one-parameter deformation, the generalized Brezin-Gross-Witten tau-function. In particular, we derive the Virasoro constraints, which completely specify the partition function. We solve them in terms of the cut-and-join operator. The Virasoro constraints lead to the loop equations, which we solve in terms of the correlation functions. Explicit expressions for the coefficients of the tau-function and the free energy are derived, and a compact formula for the genus zero contribution is conjectured.…
Let $W(z)$ be a $n\times n$ matrix polynomial with rational coefficients. Denote $C$ the spectral curve $\det \left( w\cdot{\bf 1}-W(z)\right) =0$. Under some natural assumptions about the structure of $W(z)$ we prove that certain combinations of logarithmic derivatives of the Riemann theta-function of $C$ of an arbitrary order starting from the third one all take rational values at the point of the Jacobi variety $J(C)$ specified by the line bundle of eigenvectors of $W(z)$.
We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel-Jacobi images of k points on a Riemann surface with k less than or equal to g. We determine …
We discuss some algebraic aspects of the integrable vector non-linear Schrödinger hierarchies (GNLS$_{r}$). These are hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras $\hat{sl}_{r+1}$. Using the dressing transformation method and the tau-function formalism, we construct the N-soliton solutions of the GNLS$_{r}$ systems. The explicit matrix elements in the case of GNLS$_{1}$ are computed using level one vertex operator representations.
For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten--Kontsevich tau-function, to the generalized Brézin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.
Littlewood-Richardson rule gives the decomposition formula for the multiplication of two Schur functions, while the decomposition formula for the multiplication of two Hall-Littlewood functions or two universal characters is also given by the combinatorial method. In this paper, using the vertex operator realizations of these symmetric functions, we construct the algebraic forms of these decomposition formulas.
We use the formalism of the Bergman tau functions to study the geometry of moduli spaces of holomorphic quadratic differentials on complex algebraic curves. We introduce two natural tau functions and interpret them as holomorphic sections of certain line bundles on the moduli space. Analyzing the asymptotic behavior of these tau functions near the boundary of the moduli space we get two non-trivial relation in the rational Picard group of the moduli space of quadratic differential.
