Abstract
Let (W,S) be a finite Weyl group and let w∈W. It is widely appreciated that the descent set
determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset W J where J⊆S. Our main results here include the identification of a certain subset S J⊆W J that convincingly plays the role of S⊆W, at least from the point of view of descent sets and related geometry. The point here is to use this resulting descent system (W J,S J) to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset W J. In particular, we arrive at the notion of an augmented poset, and we identify the combinatorially smooth subsets J⊆S that have special geometric significance in terms of a certain corresponding torus embedding X(J). The theory of J-irreducible monoids provides an essential tool in arriving at our main results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bialynicki-Birula, A.: Some theorems on the actions of algebraic groups. Annals of Math. 98, 480–497 (1973)
Bjorner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Brenti, F.: q-Eulerian poynomials arising from Coxeter groups. European Journal of Combinatorics 15, 417–441 (1994)
Brion, M.: Rational smoothness and fixed points of torus actions. Transformation Groups 4, 127–156 (1999)
Brown, K.: Semigroup and ring theoretical methods in probability. In: Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry. Fields Inst. Commun., vol. 40, pp. 3–26. Amer. Math. Soc., Providence (2004)
Danilov, V.I.: The geometry of toric varieties. Russian Mathematical Surveys 33, 97–154 (1978)
DeConcini, C., Procesi, C.: Complete Symmetric Varieties. Springer Lecture Notes in Mathematics, vol. 131. Springer, Berlin (1983), pp. 1–44
Dolgachev, I., Lunts, V.: A character formula for the representation of a Weyl group on the cohomology of the associated toric variety. Journal of Algebra 168, 741–772 (1994)
Grosshans, F.D.: The variety of points which are not semi-stable. Illinois Journal of Math. 26, 138–148 (1982)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, 3rd edn. GTM, vol. 9. Springer, Berlin (1980)
Putcha, M.S.: Linear Algebraic Monoids. Cambridge University Press, Cambridge (1988)
Putcha, M.S., Renner, L.E.: The system of idempotents and lattice of J-classes of reductive algebraic monoids. Journal of Algebra 116, 385–399 (1988)
Renner, L.E.: Analogue of the Bruhat decomposion for algebraic monoids. Journal of Algebra 101, 303–338 (1986)
Renner, L.E.: An explicit cell decomposition of the canonical compactification of an algebraic group. Can. Math. Bull. 46, 140–148 (2003)
Renner, L.E.: Linear Algebraic Monoids. Encyclopedia of Mathematical Sciences, vol. 134. Springer, Berlin (2005)
Stembridge, J.R.: Some permutation representations of Weyl groups associated with the cohomology of toric varieties. Advances in Mathematics 106, 244–301 (1994)
Solomon, L.: A Mackey formula in the group ring of a Coxeter group. Journal of Algebra 41, 255–264 (1976)
Solomon, L.: An introduction to reductive monoids. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups, pp. 295–352. Kluwer Academic, Dordrecht (1995)
Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. N.Y. Acad. Sci. 576, 500–534 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Renner, L.E. Descent systems for Bruhat posets. J Algebr Comb 29, 413–435 (2009). https://doi.org/10.1007/s10801-008-0141-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-008-0141-4