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Gaudin model for the multinomial distribution

Plamen Iliev

Abstract

The goal of the paper is to analyze a Gaudin model for a polynomial representation of the Kohno-Drinfeld Lie algebra associated with the multinomial distribution. The main result is the construction of an explicit basis of the space of polynomials consisting of common eigenfunctions of Gaudin operators in terms of Aomoto-Gelfand hypergeometric series. The construction shows that the polynomials in this basis are also common eigenfunctions of the operators for a dual Gaudin model acting on the degree indices, and therefore they provide a solution to a multivariate discrete bispectral problem.

Gaudin model for the multinomial distribution

Abstract

The goal of the paper is to analyze a Gaudin model for a polynomial representation of the Kohno-Drinfeld Lie algebra associated with the multinomial distribution. The main result is the construction of an explicit basis of the space of polynomials consisting of common eigenfunctions of Gaudin operators in terms of Aomoto-Gelfand hypergeometric series. The construction shows that the polynomials in this basis are also common eigenfunctions of the operators for a dual Gaudin model acting on the degree indices, and therefore they provide a solution to a multivariate discrete bispectral problem.
Paper Structure (6 sections, 3 theorems, 56 equations)

This paper contains 6 sections, 3 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Multivariate Krawtchouk polynomials
  3. Representations of the Kohno-Drinfeld algebra for the multinomial distribution
  4. Parametrization and spectral properties of multivariate Krawtchouk polynomials
  5. Algebraic properties of $L_{i,j}$ and conditions for diagonalization
  6. Diagonalization of Gaudin operators

Key Result

Lemma 3.1

The operators $\{ L_{i,j} \}_{{0\leq i< j\leq d}}$ are linearly independent as operators acting on the space $\mathbb{R}_{1}[x]=\mathrm{span}\{1,x_{1},\dots,x_{d}\}$.

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Theorem 4.1
  • proof
  • ...and 1 more