but the resulting algebra is different from the Hecke group algebra of S n . Now one has a W -action and an H W (0)-action on the Stanley-Reisner ring F[?(W )]. What can we say about the algebra generated by the operators s i and ? i on F[?(W )]? Is it the same as the Hecke group algebra of W ? If not, what properties (dimension, bases, presentation, simple and projective indecomposable modules, etc.) does it have? 7.3 Tits Building Let ?(G) be the Tits building of the general linear group G = GL(n, F q ) and its usual BN-pair over a finite field F q ; see e.g. Björner [4]. The Stanley-Reisner ring F[?(G)] is a q-analogue of F[B n ]. The nonzero monomials in F[?(G)] are indexed by multiflags of subspaces of F n q , and there are q inv(w) many multiflags corresponding to a given multichain M in B n, Can one obtain the multivariate quasisymmetric function identities in Theorem 1.2 by defining a nice H n (0)-action on F[?(G)]? References [1] R. Adin, F. Brenti, and Y. Roichman, Descent representations and multivariate statistics, pp.357-3051, 2005. ,
Elements of the representation theory of associative algebras Techniques of representation theory, 2006. ,
q and q, t-analogs of non-commutative symmetric functions, Discrete Math, pp.79-103, 2005. ,
Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Advances in Mathematics, vol.52, issue.3, pp.173-212, 1984. ,
DOI : 10.1016/0001-8708(84)90021-5
Generalized Quotients in Coxeter Groups, Transactions of the American Mathematical Society, vol.308, issue.1, pp.1-37, 1988. ,
DOI : 10.2307/2000946
A Combinatorial Property of q-Eulerian Numbers, The American Mathematical Monthly, vol.82, issue.1, pp.51-54, 1975. ,
DOI : 10.2307/2319133
Combinatorial methods in the theory of Cohen-Macaulay rings, Advances in Mathematics, vol.38, issue.3, pp.229-266, 1980. ,
DOI : 10.1016/0001-8708(80)90006-7
Permutation statistics and partitions, Advances in Mathematics, vol.31, issue.3, pp.288-305, 1979. ,
DOI : 10.1016/0001-8708(79)90046-X
On certain graded Sn-modules and the q-Kostka polynomials, Advances in Mathematics, vol.94, issue.1, pp.82-138, 1992. ,
DOI : 10.1016/0001-8708(92)90034-I
Counting permutations with given cycle structure and descent set, Journal of Combinatorial Theory, Series A, vol.64, issue.2, pp.189-215, 1993. ,
DOI : 10.1016/0097-3165(93)90095-P
A specialization theorem for certain Weyl group representations and an application to the green polynomials of unitary groups, Inventiones Mathematicae, vol.36, issue.5, pp.41-113, 1977. ,
DOI : 10.1007/BF01418371
The Hecke group algebra of a Coxeter group and its representation theory, Journal of Algebra, vol.321, issue.8, pp.321-2230, 2009. ,
DOI : 10.1016/j.jalgebra.2008.09.039
URL : https://hal.archives-ouvertes.fr/hal-00484684
0-Hecke algebra actions on coinvariants and flags, to appear in J. Algebraic Combin ,
Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q = 0, Journal of Algebraic Combinatorics, vol.6, issue.4, pp.339-376, 1997. ,
DOI : 10.1023/A:1008673127310
URL : https://hal.archives-ouvertes.fr/hal-00018539
Noncommutative symmetric functions with matrix parameters, Journal of Algebraic Combinatorics, vol.224, issue.1???3, pp.621-642, 2013. ,
DOI : 10.1007/s10801-012-0378-9
URL : https://hal.archives-ouvertes.fr/hal-00786597
Oddification of the Cohomology of Type A Springer Varieties, International Mathematics Research Notices ,
DOI : 10.1093/imrn/rnt098
Combinatorial Analysis, 1915. ,
0-Hecke algebras, Journal of the Australian Mathematical Society, vol.27, issue.03, pp.337-357, 1979. ,
DOI : 10.1016/0021-8693(68)90022-7
Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tohoku Mathematical Journal, vol.34, issue.4, pp.575-585, 1982. ,
DOI : 10.2748/tmj/1178229158