Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

TL;DR

This paper provides an algebraic framework for matrix-valued orthogonal polynomials using Lie algebra representations, leading to new matrix analogs of classical discrete polynomials.

Contribution

It introduces a novel algebraic interpretation of MVOPs via Lie algebra representations, including $q$-deformed cases, connecting to classical polynomial families.

Findings

Constructs MVOPs from Lie algebra representations.

Derives matrix analogs of Krawtchouk, Meixner, and Chebyshev polynomials.

Includes cases for $rak{su}(2)$, $rak{su}(1,1)$, and $rak{so}_q(3)$ at roots of unity.

Abstract

An algebraic interpretation of matrix-valued orthogonal polynomials (MVOPs) is provided. The construction is based on representations of a ($q$-deformed) Lie algebra $\mathfrak{g}$ into the algebra $\operatorname{End}_{M_n(\mathbb{C})}(M)$ of $M_n(\mathbb{C})$-linear maps over a $M_n(\mathbb{C})$-module $M$. Cases corresponding to the Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(1, 1)$ as well as to the $q$-deformed algebra $\mathfrak{so}_q(3)$ at $q$ a root of unity are presented; they lead to matrix analogs of the Krawtchouk, Meixner and discrete Chebyshev polynomials.