Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials

Pavel Etingof, Alexander jun. Kirillov

Publication date: 21 October 1994

Abstract: This paper is a continuation of our papers cite{EK1, EK2}. In cite{EK2} we showed that for the root system

A_{n - 1}

one can obtain Macdonald's polynomials as weighted traces of intertwining operators between certain finite-dimensional representations of

U_{q} (s l_{n})

. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system

A_{n - 1}

. The proofs are based on the techniques of ribbon graphs developed by Reshetikhin and Turaev. We also use the symmetry identities to derive recursive relations for Macdonald's polynomials.

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