Kazhdan-Lusztig polynomials and canonical basis

Frenkel, Igor; Khovanov, Mikhail; Kirillov Jr, Alexander

Kazhdan-Lusztig polynomials and canonical basis

Authors

Frenkel, Igor

Khovanov, Mikhail

Kirillov Jr, Alexander

Description

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.
15 pages, AMSTeX, no figures