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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