×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Bernstein, I.N. , Gelfand, I.M. and Gelfand, S.I. , Structure of representations generated by highest weight vectors (English translation) Funct. Anal. Appl. 5 (1971) 1-8. · Zbl 0246.17008
[2] Bjorner, A. , Orderings of Coxeter groups , in ” Combinatorics and algebra (Boulder, Colo. 1983)”, Contemp Math 34, Amer. Math. Soc., Providence, R. I (1984). · Zbl 0594.20029
[3] Bjorner, A. and Wachs, M. , Bruhat order of Coxeter groups and shellability , Adv. in Math. 43 (1982), 87-100. · Zbl 0481.06002
[4] Carter, R.W. , Finite groups of Lie type , John Wiley & Sons, Chichester (1985). · Zbl 0567.20023
[5] Davis, M.W. , Groups generated by reflections and aspherical manifolds not covered by Euclidean space , Ann. of Math. 117(2) (1983) 293-324. · Zbl 0531.57041
[6] Deodhar, V. , On the root system of a Coxeter group , Comm. Alg. 10 (1982) 611-630. · Zbl 0491.20032
[7] Dyer, M. , Hecke algebras and reflections in Coxeter groups , Ph.D. Thesis, University of Sydney (1987).
[8] Dyer, M. , Reflection subgroups of Coxeter systems , to appear in J. of Alg. · Zbl 0712.20026
[9] Dyer, M. , Hecke algebras and shellings of Bruhat intervals , preprint. · Zbl 0817.20045
[10] Dyer, M. , Hecke algebras and shellings of Bruhat intervals II; twisted Bruhat orders , preprint. · Zbl 0817.20045
[11] Jantzen, J.C. , Representations of algebraic groups , Pure and Applied Math. Vol. 131, Academic Press, Boston (1987). · Zbl 0654.20039
[12] Kazhdan, D. and Lusztig, G. , Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979), 165-184. · Zbl 0499.20035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.