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The left cells of the Coxeter group of type \(H_ 4\). (English) Zbl 0615.20019

Let W be the Coxeter group of type \(H_ 4\). The author gives an explicit description for the left and two-sided cells of W as sets. He also establishes several properties of these cells, these properties were proved in certain crystallographic cases by Lusztig. The proofs of all the above results are based on the calculation of all the Kazhdan-Lusztig polynomials \(P_{y,w}\), y, \(w\in W\), by a computer.
Reviewer: Shi Jianyi

MSC:

20G05 Representation theory for linear algebraic groups
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References:

[1] Alvis, D.; Lusztig, G., The representations and generic degrees of the Hecke algebra of type \(H_4\), J. Reine Angew. Math., 336, 201-212 (1982) · Zbl 0488.20034
[2] Grove, L. C., The characters of the hecatonicosahedroidal group, J. Reine Angew. Math., 265, 160-169 (1974) · Zbl 0275.20015
[3] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184 (1979) · Zbl 0499.20035
[4] Lusztig, G., On a theorem of Benson and Curtis, J. Algebra, 71, 490-498 (1981) · Zbl 0465.20042
[5] Lusztig, G., A class of irreducible representations of a Weyl group II, (Proc. Kon. Nederl. Akad. Ser. A, 85 (1982)), 219-226 · Zbl 0511.20034
[6] Lusztig, G., Characters of Reductive Groups over a Finite Field, (Ann. of Math. Studies, Vol. 103 (1984), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J) · Zbl 0930.20041
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