Data on Kazhdan-Lusztig polynomials up to S_11 suggests superexponential growth of extremal coefficients, near-universal unimodality, and a conjectured closed family (1+v+...+v^l)^{k-1}.
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 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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Big data approach to Kazhdan-Lusztig polynomials
Data on Kazhdan-Lusztig polynomials up to S_11 suggests superexponential growth of extremal coefficients, near-universal unimodality, and a conjectured closed family (1+v+...+v^l)^{k-1}.