LU-decomposition of a noncommutative linear system and Jacobi polynomials

Details LU-decomposition of a noncommutative linear system and Jacobi polynomials LU-decomposition of a noncommutative linear system and Jacobi polynomials

2008-10-15

arxiv.org/abs/0810.2771v1

Mathematics

Representation Theory

Scientific paper

In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the ultraspherical Jacobi polynomials.

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