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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. Björner [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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