Algebraic and combinatorial properties of zircons
Mario Marietti
Università di Roma “La Sapienza” Dipartimento di Matematica Piazzale Aldo Moro 5 00185 Roma Italy
DOI: 10.1007/s10801-007-0061-8
Abstract
In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the context of zircons. Moreover, for any zircon Z, we construct and count all balanced and exact labelings (used in the construction of the Bernstein-Gelfand-Gelfand resolutions in the case that Z is the Weyl group of a Kac-Moody algebra).
Pages: 363–382
Keywords: keywords Bruhat order; special matchings; Coxeter groups
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References
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231. Springer, New York (2005)
3. Björner, A., Wachs, M.: Bruhat order of Coxeter groups and shellability. Adv. Math. 43, 87-100 (1982)
4. Björner, A., Wachs, M.: On lexicographically shellable posets. Trans. Am. Math. Soc. 277, 323-341 (1983)
5. Bourbaki, N.: Groupes et Algèbres de Lie, Chaps. 4-6. Hermann, Paris (1968)
6. Brenti, F.: The intersection cohomology of Schubert varieties is a combinatorial invariant. Eur. J. Comb. 25, 1151-1167 (2004)
7. Brenti, F., Caselli, F., Marietti, M.: Special matchings and Kazhdan-Lusztig polynomials. Adv. Math. 202, 555-601 (2006)
8. Brenti, F., Caselli, F., Marietti, M.: Diamonds and Hecke algebra representations. Int. Math. Res. Not. 2006, 34 (2006), article ID 29407
9. Deodhar, V.V.: Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function. Invent. Math. 39, 187-198 (1977)
10. Dyer, M.J.: Hecke algebras and reflections in Coxeter groups. Ph.D. thesis, University of Sydney (1987) 11. du Cloux, F.: An abstract model for Bruhat intervals. Eur. J. Comb. 21, 197-222 (2000)
12. Hiller, H.: Geometry of Coxeter Groups. Research Notes in Mathematics, vol.
54. Pitman Advanced Publishing Programm (1982)
13. Hultman, A.: The combinatorics of twisted involutions in Coxeter Groups. Trans. Am. Math. Soc. 359, 2787-2798 (2007)
14. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol.
29. Cambridge Univ. Press, Cambridge (1990)
15. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165-184 (1979)
16. Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, vol.
204. Birkhäuser, Basel (2002)
17. Marietti, M.: Kazhdan-Lusztig theory: Boolean elements, special matchings and combinatorial invariance. Ph.D. Thesis, Università degli Studi di Roma “La Sapienza”, Italy (2003)
18. Stembridge, J.: A short derivation of the Möbius function for the Bruhat order. J. Algebr. Comb. 25(2), 141-148 (2007)
19. Verma, D.-N.: Structure of certain induced representations of complex semisimple Lie algebras. Bull.