A Hecke Algebra Quotient and Some Combinatorial Applications
C.K. Fan
DOI: 10.1023/A:1022443327568
Abstract
Let ( W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient _boxclose_boxclose \bar H of H and show that it has a basis parametrized by a certain subset W c of the Coxeter group W. Specifically, W c consists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W c and compute the cardinality of W c when W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for [ `( H)] \bar H .
Pages: 175–189
Keywords: permutation; representation theory; non-commutative algebra; Lie theory; reductive group
Full Text: PDF
References
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2. S.C. Billey, W. Jockusch, and R.P. Stanley, "Some combinatorial properties of Schubert polynomials," J. Alg. Combin. 2 (1993), 345-374.
3. N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5, et 6, Masson, Paris, 1981.
4. C.K. Fan and G. Lusztig, "Factorization of certain exponentials in Lie groups," to appear in Tribute to R.W. Richardson.
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. V.F.R. Jones, "A polynomial invariant for knots via Von Neumann algebras," Bulletin of the Amer. Math. Soc. 12(1) (1985), 103-111.
7. V.F.R. Jones, "Hecke algebra representations of braid groups and link polynomials," Annals of Mathematics 126(1987), 335-388.
8. G. Lusztig, "Characters of reductive groups over a finite field," Annals of Mathematical Studies No. 107, Princeton University Press, 1984.
9. G. Lusztig, "Total positivity in reductive groups," in Lie Theory and Geometry (In Honor of Bertram Kostant), BirkhaUser, Boston, 1994.
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.