Kazhdan-Lusztig polynomials and canonical basis

Details Kazhdan-Lusztig polynomials and canonical basis Kazhdan-Lusztig polynomials and canonical basis

1997-09-29

arxiv.org/abs/q-alg/9709042v1

Transform. Groups 3 (1998), no. 4, 321--336

Mathematics

Quantum Algebra

15 pages, AMSTeX, no figures

Scientific paper

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 $U_q sl_k$. We also use known results about canonical bases for $U_q sl_2$ 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.

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