Canonical Basis and Macdonald Polynomials

1998/06/28 by Jonathan Beck, Igor Frenkel, Naihuan Jing
Mathematics · #math.QA #math.CO #msc:17B #msc:05

paper · pdf

published as Advances in Math. 140 (1998), 95-127. · 25 pages, Latex2e. Advances in Math, to appear

arxiv created 1998/06/28 · arxiv updated 2009/11/30

Abstract

In the basic representation of U_q(sl(2)) realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with q=t². We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also discuss the positivity and integrality of these coefficients. For level k, we expect a similar relation between the canonical basis and Macdonald polynomials with q²=t^k.

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