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 image, 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 image of the Hecke algebra Hn(q). Using the immanant and factorization results, we show that for every totally nonnegative immanant image and its expansion image with respect to the basis of Kazhdan–Lusztig immanants, the coefficient dw must be nonnegative when w avoids the patterns 3412 and 4231.