On the dual canonical and Kazhdan–Lusztig bases and 3412-, 4231-avoiding permutations

Abstract

Using Du’s characterization of the dual canonical basis of the coordinate ring O ( G L ( n , C ) ) , we express all elements of this basis in terms of immanants. We then give a new factorization of permutations w avoiding the patterns 3412 and 4231, which in turn yields a factorization of the corresponding Kazhdan–Lusztig basis elements C w ′ ( q ) of the Hecke algebra H n ( q ) . Using the immanant and factorization results, we show that for every totally nonnegative immanant Imm f ( x ) and its expansion ∑ d w Imm w ( x ) with respect to the basis of Kazhdan–Lusztig immanants, the coefficient d w must be nonnegative when w avoids the patterns 3412 and 4231.