Algebraic and combinatorial properties of zircons

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). Partially supported by the program “Gruppi di trasformazioni e applicazioni”, University of Rome “La Sapienza”. Part of this research was carried out while the author was a member of the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences.