Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 084, 28 pages      arXiv:2002.02148      https://doi.org/10.3842/SIGMA.2020.084
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions

Ayumu Hoshino ^a and Jun'ichi Shiraishi ^b
a)  Hiroshima Institute of Technology, Miyake, Hiroshima 731-5193, Japan
^b)  Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received April 02, 2020, in final form August 18, 2020; Published online August 25, 2020

Abstract
We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $BC_n$ Koornwinder polynomials $P_{(1^r)}(x|a,b,c,d|q,t)$ with one column diagrams, to the type $BC_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from $BC_n$ to $BC_{n-1}$. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: $P_{(1^r)}(x|a,b,c,d|q,t) \longleftrightarrow P_{(1^r)}(x|a,-a,c,d|q,t)\longleftrightarrow P_{(1^r)}(x|a,-a,c,-c|q,t) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2}c,-t^{1/2}c,c,-c|q,t\big) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2},-t^{1/2},1,-1|q,t\big)$. We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type $B_n$, namely the transition matrix from the Schur polynomials $P^{(B_n,B_n)}_{(1^r)}(x|q;q,q)$ to the Hall-Littlewood polynomials $P^{(B_n,B_n)}_{(1^r)}(x|t;0,t)$. We also present a conjecture for the asymptotically free eigenfunctions of the $B_n$ $q$-Toda operator, which can be regarded as a branching formula from the $B_n$ $q$-Toda eigenfunction restricted to the $A_{n-1}$ $q$-Toda eigenfunctions.

Key words: Koornwinder polynomial; degeneration scheme; Kostka polynomial of type $B_n$; $q$-Toda eigenfunction.

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