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Séminaire Lotharingien de Combinatoire, B66b (2011), 68 pp.

# Charles F. Dunkl and Jean-Gabriel Luque

# Vector Valued Macdonald Polynomials

**Abstract.**
This paper defines and investigates nonsymmetric Macdonald polynomials
with values in an irreducible module of the Hecke algebra of type
*A*_{N-1}. These polynomials appear as simultaneous eigenfunctions
of Cherednik operators. Several objects and properties are
analyzed, such as the canonical bilinear form which pairs
polynomials with those arising from reciprocals of the original
parameters, and the symmetrization of the Macdonald
polynomials. The main tool of the study is the Yang-Baxter
graph. We show that these Macdonald polynomials can be easily
computed following this graph. We give also an interpretation of
the symmetrization and the bilinear forms applied to the Macdonald
polynomials in terms of the Yang-Baxter graph.

Received: June 25, 2011.
Revised: January 13, 2012; February 7, 2012.
Accepted: January 13, 2012.

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