Total Nonnegativity and Induced Sign Characters of the Hecke Algebra


Let \({\mathfrak {S}}_{[i,j]}\) be the subgroup of the symmetric group \({\mathfrak {S}}_n\) generated by adjacent transpositions \((i,i+1), \dotsc , (j-1,j)\), 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 \(C'_w(q)\) of the Kazhdan–Lusztig basis.

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Clearwater, A., Skandera, M. Total Nonnegativity and Induced Sign Characters of the Hecke Algebra. Ann. Comb. (2021).

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