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Hecke algebras and shellings of Bruhat intervals. (English) Zbl 0817.20045

Let \((W,S)\) be a Coxeter system. The aim of the present paper is to study the connections between the Iwahori-Hecke algebra \({\mathcal H}(W)\) of \((W,S)\) and the combinatorial structure of Bruhat intervals. The author describes the Hecke algebra \({\mathcal H}(W)\) as an algebra of functions under twisted convolution product. Then he defines so called interpolating polynomials for the KL-polynomials and for the inverse KL- polynomials to be the values assumed by the functions corresponding to certain Kazhdan-Lusztig basis elements. Hence the author obtains new definitions (which are equivalent to the original ones) of the KL- polynomials and the inverse KL-polynomials. He uses these new definitions to discuss the non-negativity of the coefficients of the KL-polynomials and the inverse KL-polynomials. On the other hand, the author defines a total order, called a reflection order, on the set of all the reflections of \(W\). He shows that reflection orders give rise to KL-shellings of Bruhat intervals.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G05 Representation theory for linear algebraic groups
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References:

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