A combinatorial setting for questions in Kazhdan — Lusztig theory
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Let(W, S) be a Coxeter group. Let s1 ... sk be a reduced expression for an element y in W. A combinatorial setting involving subexpressions of this reduced expression is developed. This leads to the notion of good elements. It is proved that all elements in a group where the coefficients of Kazhdan — Lusztig polynomials are non-negative are good. If y is good then an algorithm is developed to compute these polynomials in a very efficient way. It is further proved that in these cases, the coefficients of these polynomials can be identified as sizes of certain subsets of subexpressions thereby providing an explicit setting for various questions regarding these polynomials and related topics. Similar results are obtained for the so-called parabolic case.
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