Reducibility of Matrix Weights

TL;DR

This paper explores the concept of reducibility in matrix weights, introducing a vector space framework, and shows how reducibility relates to commutants, orthogonal polynomials, and differential operators, with implications for polynomial reducibility.

Contribution

It introduces a new vector space framework for analyzing reducibility of matrix weights and links reducibility to algebraic structures like commutants and differential operators.

Findings

A matrix weight reduces iff a non-scalar T satisfies TW=WT*

Reducibility can be studied via orthogonal polynomial commutants

Any matrix weight is equivalent to a direct sum of irreducible weights

Abstract

In this paper we discuss the notion of reducibility for matrix weights and introduce a real vector space $\mathcal C_\mathbb{R}$ which encodes all information about the reducibility of $W$. In particular a weight $W$ reduces if and only if there is a non-scalar matrix $T$ such that $TW=WT^*$. Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent. We consider the algebra of right-hand-side matrix differential operators $\mathcal D(W)$ of a reducible weight $W$, giving its general structure. Finally, we make a change of emphasis by considering reducibility of polynomials, instead of reducibility of matrix weights.