Total nonnegativity and induced sign characters of the Hecke algebra

Abstract

Let ${\mathfrak{S}}_{[i,j]}$ be the subgroup of the symmetric group ${\mathfrak{S}}_n$ generated by adjacent transpositions $(i,i+1),\dots ,(j-1,j)$, assuming $1\le i<j\le n$. 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 ${\sum }_{w\in {\mathfrak{S}}_{[i,j]}}{T}_w$ and at all products of such elements. This includes evaluation at some elements $C_w^{\prime }(q)$ of the Kazhdan-Lusztig basis.

24 pages