# zbMATH — the first resource for mathematics

Generalized Jones traces and Kazhdan-Lusztig bases. (English) Zbl 1172.20004
Summary: We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the positivity of certain structure constants for the associated Kazhdan-Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading terms of certain Kazhdan-Lusztig polynomials.

##### MSC:
 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text:
##### References:
 [1] Björner, A.; Brenti, F., Combinatorics of Coxeter groups, (2005), Springer New York · Zbl 1110.05001 [2] Brenti, F., A combinatorial formula for kazhdan – lusztig polynomials, Invent. math., 118, 371-394, (1994) · Zbl 0836.20054 [3] Tom Dieck, T., Bridges with pillars: A graphical calculus of knot algebra, Topology appl., 78, 21-38, (1997) · Zbl 0879.57002 [4] Fan, C.K., Structure of a Hecke algebra quotient, J. amer. math. soc., 10, 139-167, (1997) · Zbl 0861.20042 [5] Geck, M., Kazhdan – lusztig cells and the murphy basis, Proc. London math. soc., 93, 635-665, (2006) · Zbl 1158.20001 [6] Geck, M.; Pfeiffer, G., Characters of finite Coxeter groups and iwahori – hecke algebras, (2000), Oxford University Press Oxford · Zbl 0996.20004 [7] J.J. Graham, Modular representations of Hecke algebras and related algebras, Ph.D. Thesis, University of Sydney, 1995 [8] Green, R.M., Generalized temperley – lieb algebras and decorated tangles, J. knot theory ramifications, 7, 155-171, (1998) · Zbl 0926.20005 [9] Green, R.M., Tabular algebras and their asymptotic versions, J. algebra, 252, 27-64, (2002) · Zbl 1062.16030 [10] Green, R.M., On planar algebras arising from hypergroups, J. algebra, 263, 126-150, (2003) · Zbl 1064.20007 [11] Green, R.M., Star reducible Coxeter groups, Glasgow math. J., 48, 583-609, (2006) · Zbl 1149.20034 [12] R.M. Green, On the Markov trace for Temperley-Lieb algebras of type $$E_n$$, Preprint available on arXiv:0704.0283 [13] Green, R.M.; Losonczy, J., Canonical bases for Hecke algebra quotients, Math. res. lett., 6, 213-222, (1999) · Zbl 0961.20007 [14] Green, R.M.; Losonczy, J., A projection property for kazhdan – lusztig bases, Int. math. res. not., 1, 23-34, (2000) · Zbl 0961.20008 [15] Green, R.M.; Losonczy, J., Fully commutative kazhdan – lusztig cells, Ann. inst. Fourier (Grenoble), 51, 1025-1045, (2001) · Zbl 1008.20036 [16] Humphreys, J.E., Reflection groups and Coxeter groups, (1990), Cambridge University Press Cambridge · Zbl 0725.20028 [17] Jones, V.F.R., Hecke algebra representations of braid groups and link polynomials, Ann. of math., 126, 2, 335-388, (1987) · Zbl 0631.57005 [18] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. math., 53, 165-184, (1979) · Zbl 0499.20035 [19] Losonczy, J., The kazhdan – lusztig basis and the temperley – lieb quotient in type D, J. algebra, 233, 1-15, (2000) · Zbl 0969.20003 [20] Lusztig, G., Cells in affine Weyl groups, (), 255-287 [21] Lusztig, G., Introduction to quantum groups, (1993), Birkhäuser Basel · Zbl 0788.17010 [22] Shi, J.Y., Fully commutative elements and kazhdan – lusztig cells in the finite and affine Coxeter groups, Proc. amer. math. soc., 131, 3371-3378, (2003) · Zbl 1064.20043 [23] Shi, J.Y., Fully commutative elements and kazhdan – lusztig cells in the finite and affine Coxeter groups, II, Proc. amer. math. soc., 133, 2525-2531, (2005) · Zbl 1072.20048 [24] Shi, J.Y., Fully commutative elements in the Weyl and affine Weyl groups, J. algebra, 284, 13-36, (2005) · Zbl 1079.20059 [25] Stembridge, J.R., On the fully commutative elements of Coxeter groups, J. algebraic combin., 5, 353-385, (1996) · Zbl 0864.20025
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.