Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
Quentin Labriet, Lucia Morey, Luc Vinet
TL;DR
The paper develops an algebraic framework to realize matrix-valued orthogonal polynomials (MVOPs) as transition coefficients between eigenbases generated by two self-adjoint operators within Morita-equivalent module settings. By representing a (q-deformed) Lie algebra on $M_n(\mathbb{C})$-modules and enforcing a tridiagonal action, MVOPs inherit three-term recurrence, orthogonality with a matrix weight, and associated difference or differential equations. The authors instantiate this program for three algebras: $\mathfrak{su}(2)$ (yielding Krawtchouk-type MVOPs), $\mathfrak{su}(1,1)$ (Meixner-type MVOPs), and $\mathfrak{so}_q(3)$ at a root of unity (discrete Chebyshev-type MVOPs), including explicit $n\times n$ and $2\times 2$ constructions and their weights. The approach unifies scalar families with matrix-valued analogues and provides concrete formulas for weights, recurrences, and difference equations, opening avenues for classification and applications in mathematical physics.
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.