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Reducibility of matrix weights

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Abstract

In this paper, we discuss the notion of reducibility of 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 nonscalar 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 the reducibility of polynomials, instead of reducibility of matrix weights.

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Authors and Affiliations

  1. FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba, 5000, Argentina

    Juan Tirao

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, 7820436, Chile

    Ignacio Zurrián

Authors
  1. Juan Tirao
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  2. Ignacio Zurrián
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Correspondence to Ignacio Zurrián.

Additional information

This paper was partially supported by CONICET PIP 112-200801-01533 and by the program Oberwolfach Leibniz Fellows 2015.

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Tirao, J., Zurrián, I. Reducibility of matrix weights. Ramanujan J 45, 349–374 (2018). https://doi.org/10.1007/s11139-016-9834-9

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