, Volume 3, Issue 4, pp 321-336

Kazhdan-Lusztig polynomials and canonical basis

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Abstract

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the groupS n coincide with the coefficients of the canonical basis innth tensor power of the fundamental representation of the quantum groupU q \(\mathfrak{s}\mathfrak{l}\) k . We also use known results about canonical bases forU q \(\mathfrak{s}\mathfrak{l}\) 2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.