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Register a new userDrinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
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Ann du Crest de Villeneuve
Publication date
2017
fields
Physics
and research's language is
English
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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 Plucker 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.
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Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble. Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Pade approximants. This yields the first example regarding the connection between integrable lattices and vector Pade approximants, for which Hietarinta, Joshi and Nijhoff pointed out that This field remains largely to be explored. in the recent monograph [27, Section 4.4] .
We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painleve VI tau function. The result proves the conjectural formula for this constant proposed in cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painleve II tau function.
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The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a realization of the recoupling problem in terms of three Dunkl oscillators.
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