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On the “Bruhat graph” of a Coxeter system. (English) Zbl 0784.20019
Let \((W,R)\) be a finite Coxeter system. The author introduces the “Bruhat graph” associated with \((W,R)\) which determines the Bruhat order, but exhibits a kind of functoriality with respect to inclusions of reflection subgroups not shown by the Bruhat order. Using the “Bruhat graph” associated with \((W,R)\), the author answers a question of A. Bj√∂rner [Contemp. Math. 34, 175-195 (1984; Zbl 0594.20029)] by showing that only finitely many isomorphism types of posets of fixed length \(n\) occur as Bruhat intervals in \((W,R)\) and shows that the pairs of elements \(x\), \(y\) of \(W\) such that \(x^{-1}y\) is a reflection are determined by the Bruhat order as an abstract order. This supports the conjecture that the Kazhdan-Lusztig polynomial \(P_{x,y}\) [D. Kazhdan and G. Lusztig, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] depends only on the isomorphism type of the Bruhat interval \([x,y]\).

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G05 Representation theory for linear algebraic groups
06A07 Combinatorics of partially ordered sets
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References:
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