We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the pos-itivity of certain structure constants for the associated Kazhdan–Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading ter...

We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e, u] only depend on the isomorphism type of [e, u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for sp...

We determine the set of connected components of minuscule affine Deligne-Lusztig varieties for special maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne-Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected com...

We establish a formalism for working with incidence algebras of posets symmetries, and we develop equivariant Kazhdan–Lusztig–Stanley theory within this formalism. This gives new way thinking about the Kazhdan–Lusztig polynomial Z-polynomial matroid.

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusz...

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli–Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form. 2009 Els...

Let G be a finite group acting on a symplectic complex vector space V . Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by “symplectic reflections”, i.e. symplectomorphisms with fixed space of codimension 2 in V . Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakam...

To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category AFF(C)κ of smooth modules (in the sense of Kazhdan and Lusztig [9]) of finite length over the corresponding affine Kac– Moody algebra in the case of central charge less than the critical level. Equivalent characterizations ...

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank n+1 thagomizer matroid by showing that the coefficient of tk is equal to the number of Dyck paths of semilength n with k long ascents. We also give a conjecture for the Sn-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.

We give an informal introduction to the theory of Deligne-Lusztig which gives all the irreducible representations of reductive groups over finite fields [DL], with an emphasis on the geometry of the Deligne-Lusztig variety. Disclaimer: by no means complete! Comments welcome / use at your own risk!