Kazhdan-Lusztig polynomials and canonical basis

Abstract

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_{q}s{l}_k$. We also use known results about canonical bases for $U_{q}s{l}_2$ to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmanians), due to Lascoux-Schutzenberger and Zelevinsky.

arXiv comment15 pages, AMSTeX, no figures

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