Speaker: Mark Skandera (Lehigh University)
Abstract: The Kazhdan-Lusztig basis {C'_w | w in S_n} of the Hecke algebra H_n is related to the natural basis {T_v | v in S_n} of H_n by a matrix whose entries are recursively-defined polynomials {P_{v,w}(q) | v,w in S_n} in N[q] known as the Kazhdan-Lusztig polynomials. No known combinatorial formula interprets the coefficients of these polynomials as set cardinalities. Nevertheless, some results which depend upon pattern avoidance in the permutation w permit one to factor the Kazhdan-Lusztig basis element C'_w in a way which provides combinatorial formulas for coefficients of the polynomials {P_{v,w}(q) | v in S_n} having second index w. No characterization of the permutations w permitting such a factorization is known. We present a negative result: conditions on w which imply that such a factorization does not exist.