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On some generalizations of the Kazhdan-Lusztig polynomials for “universal” Coxeter systems. (English) Zbl 0646.20037

The Coxeter group (W,R) is said to be universal if for any r,s\(\in R\) with \(r\neq s\), the order of rs is infinite. Let H(W) be the Hecke algebra of such W over the ring \(A={\mathbb{Z}}[q^{1/2},q^{-}]\), where \(q^{1/2}\) is an indeterminate over \({\mathbb{Z}}\). Then H(W) has two A-bases \(\{T_ w\}_{w\in W}\) and \(\{C_ w\}_{w\in W}\) defined by D. Kazhdan and G. Lusztig [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. In this paper, the author introduces two polynomials \(P^ y_{x,w}\) and \(Q^ y_{x,w}\) in \({\mathbb{Z}}[q]\) for x,y,z\(\in W\), defined by \[ T_ x+C_ w=\epsilon_ x\epsilon_ wq^{1/2}\sum_{y\in W}\epsilon_ y\bar P^ y_{x,w}T_{y^{-1}}\quad and\quad T_ wC_ y=q_ wq^{1/2}_ y\sum_{x\in W}q_ x^{-}\bar Q^ y_{x,w}C_ x, \] where for any \(w\in W\), \(\epsilon_ w=(-1)^{\ell (w)}\), \(q_ w=q^{\ell (w)}\), \(\ell (w)\) is the length of w, and \(a\mapsto \bar a\) is an involution of the ring A satisfying \(\overline{\sum_{i\in {\mathbb{Z}}}a_ iq^{i/2}}=\sum_{i\in {\mathbb{Z}}}a_ iq^{-}\), \(a_ i\in {\mathbb{Z}}\). These two polynomials can also be defined for any Coxeter group and they are a generalization of the Kazhdan-Lusztig polynomials \(P_{x,w}\) and \(Q_{x,w}\) [loc. cit], as we have \(P_{x,w}=P^ 1_{x,w}\) and \(Q_{x,w}=Q^ 1_{x,w}\). For x,y,w\(\in W\), let \(h_{x,y,w}=\sum_{n\in {\mathbb{Z}}}(-1)^ na_{w,n}q^{n/2}\in A\) with \(a_{w,n}\in {\mathbb{Z}}\) be defined by \(C_ xC_ y=\sum_{w}h_{x,y,w}C_ w.\)
Then the main results of this paper are explicit combinatorial formulae for \(P^ y_{x,w}\), \(Q^ y_{x,w}\) and \(h_{x,y,w}\). From these formulae, the author points out that for all x,y,w\(\in W\), the polynomial \(P^ y_{x,w}\) has non-negative coefficients and \(Q^ y_{x,w}\) has all its coefficients of the same sign, and that the integers \(a_{w,n}\) are all non-negative.
Reviewer: Shi Jianyi

MSC:

20H15 Other geometric groups, including crystallographic groups

Citations:

Zbl 0499.20035
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References:

[2] Bourbaki, N., Groupes et algèbres de Lie (1968), Hermann: Hermann Paris, Chaps. 4-6
[3] Deodhar, V., Some characterisations of Bruhat ordering on a Coxeter group and determination of the relative Mobius function, Invent. Math, 39, 187-198 (1977) · Zbl 0333.20041
[4] Dyer, M., Hecke algebras and reflections in Coxeter groups, (Ph.D. thesis (August 1987), University of Sydney)
[5] Kazdhan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184 (1979) · Zbl 0499.20035
[6] Kazdhan, D.; Lusztig, G., Schubert Varieties and Poincare Duality, (Proceedings Symp. Pure Math., Vol. 36 (1980), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 185-203
[7] Lusztig, G., Cells in affine Weyl groups, (Algebraic groups and related topics. Algebraic groups and related topics, Advanced Studies in Pure Math., Vol. 6 (1985), Kinokunia and North: Kinokunia and North Holland), 255-287
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