Total nonnegativity and induced sign characters of the Hecke algebra

Abstract

Let S [i,j] be the subgroup of the symmetric group S n generated by adjacent transpositions (i, i + 1), . . . , (j − 1, j), assuming 1 ≤ i < j ≤ n. We give a combinatorial rule for evaluating induced sign characters of the type A Hecke algebra H n (q) at all elements of the form w∈S [i,j] T w and at all products of such elements. This includes evaluation at some elements C ′ w (q) of the Kazhdan-Lusztig basis.1 2 for pairs (z h 1 ,p,k , z h 2 ,p,k ) of variables sharing second and third indices,z 3,2,2 z 2,2,2 = q 1 2 z 2,2,2 z 3,2,2 ,