zbMATH — the first resource for mathematics

A Hecke algebra quotient and some combinatorial applications. (English) Zbl 0853.20028
Let \(W\) be a Coxeter group associated to a Coxeter graph which has no multiple bonds, and \(H\) be the corresponding Hecke algebra. The author defines a certain quotient \(\widetilde H\) of \(H\) and shows that it has a basis parametrized by a certain subset \(W_c\) of the Coxeter group \(W\), then determines which Coxeter groups have finite \(W_c\) and computes the cardinality of \(W_c\) when \(W\) is a Weyl group. Finally, he gives a combinatorial application of an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \(\widetilde H\).

20F55 Reflection and Coxeter groups (group-theoretic aspects)
05A19 Combinatorial identities, bijective combinatorics
Full Text: DOI
[1] D. Alvis,“Induce/restrict matrices for exceptional Weyl groups,”Manuscript.
[2] Billey, S. C.; Jockusch, W.; Stanley, R. P., Some combinatorial properties of Schubert polynomials, J.Alg. Combin., 2, 345-374, (1993) · Zbl 0790.05093
[3] N. Bourbaki, Groupes et algebres de Lie, Chapitres 4,5,et 6, Masson, Paris,1981. · Zbl 0483.22001
[4] C.K. Fan and G. Lusztig,“Factorization of certain exponentials in Lie groups,”to appear in Tribute to R.W.Richardson. · Zbl 0872.22004
[5] J.S. Frame,“The characters of the Weyl group Eg,”Computational Problems in Abstract Algebra (Oxford conference,1967),ed.J. Leech,111-130.
[6] Jones, V. F.R., A polynomial invariant for knots via Von Neumann algebras, Bulletin of the Amer.Math.Soc., 12, 103-111, (1985) · Zbl 0564.57006
[7] Jones, V. F.R., Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics, 126, 335-388, (1987) · Zbl 0631.57005
[8] G. Lusztig,“Characters of reductive groups over a finite field,”Annals of Mathematical Studies No.107, Princeton University Press,1984. · Zbl 0556.20033
[9] Lusztig, G., Total positivity in reductive groups, (1994), Boston · Zbl 0845.20034
[10] J.R. Stembridge,“On the fully commutative elements of Coxeter groups,”to appear in Journal Alg.Combin.
[11] H.N.V. Temperley and E.H. Lieb,“Relations between the percolation and colouring problem...,”Proceedings of the Royal Society of London (1971),251-280. · Zbl 0211.56703
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.