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.