Séminaire Lotharingien de Combinatoire, B80a (2019), 26 pp.

Charles F. Dunkl

A Positive-Definite Inner Product for Vector-Valued Macdonald Polynomials

Abstract. In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group SN. The polynomials are parametrized by (q,t) and are simultaneous eigenfunctions of a commuting set of Cherednik operators, which were studied by Baker and Forrester (IMRN 1997). In the Dunkl-Luque paper there is a construction of a pairing between (q-1,t-1)-polynomials and (q,t)-polynomials, and for which the Macdonald polynomials form a biorthogonal set. The present work is a sequel with the purpose of constructing a symmetric bilinear form for which the Macdonald polynomials form an orthogonal basis and of determining the region of (q,t)-values for which the form is positive-definite. Irreducible representations of the Hecke algebra are characterized by partitions of N. The positivity region depends only on the maximum hook-length of the Ferrers diagram of the partition.

Received: September 26, 2018. Revised: January 18, 2019. Accepted: January 20, 2019.

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