On the Dual Canonical and Kazhdan-lusztig Bases and 3412, 4231-avoiding Permutations

Using Du’s characterization of the dual canonical basis of the coordinate ring O(GL(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 Hn(q). Using the immanant and factorization results, we show that for every totally nonnegative immanant Immf (x) and its expansion ∑ dwImmw(x) with respect to the basis of Kazhdan-Lusztig immanants, the coefficient dw must be nonnegative when w avoids the patterns 3412 and 4231.