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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 *S*_{N}. 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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