In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced concept of matroids. It is known that both matroid can be considered as special cases Kazhdan-Lusztig-Stanley locally finite posets. framework polynomials, we study inverse functions define We also compute these boolean matroids uniform As an unexpected application obtain a ne...

Let H be the Iwahori–Hecke algebra associated with Sn, the symmetric group on n symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. While the former admits a deep geometric interpretation, the latter leads to a purely combinatorial construction of the representations of H, including the Dipper–James theory of Specht modules. In this paper, we establish...

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 c...

We define " star reducible " Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan–Lusztig bases of these groups have a nice projection property to the Temperley–Lieb type quotient, an...

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with KazhdanLusztig polynomial Pid,w(q) = 1 + q for some h. Résumé. On démontre la conjecture de Billey et Braden sur les permutations w pour lesquelles le polynôme de Kazhdan-Lusztig Pid,w(q) = 1...

A type of directed multigraph called aW -digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of Lusztig, a complete characterization of W -digraphs is given in terms of subdigraphs for dihedral parabolic subgroups. In addition, results are obtained relati...

Abstract. The unipotent variety of a reductive algebraic group G plays an important role in the representation theory. In this paper, we will consider the closure Ū of the unipotent variety in the De Concini-Procesi compactification Ḡ of a quasisimple, adjoint group G. We will prove that Ū − U is a union of some G-stable pieces introduced by Lusztig in [L4]. This was first conjectured by Luszti...

We prove that Calogero–Moser cells coincide with Kazhdan–Lusztig for dihedral groups in the equal parameter case.

In examining the relationship between the number of points over Fp on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the p-...

In Stanley [16], the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley’s positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results in Goulden and Rattan [7]. We shall see that the...