We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

One of Erdős’ favourite conjectures was that any triangle-free graph G on n vertices should contain a set of n/2 vertices that spans at most n2/50 edges. We prove this when the number of edges in G is either at most n2/12 or at least n2/5. © 2005 Elsevier Inc. All rights reserved.

Let G be a group with Kazhdan’s property (T), and let S be a transitive generating set (there exists a group H ⊂ Aut(G) which acts transitively on S.) In this paper we relate two definitions of the Kazhdan constant and the eigenvalue gap in this case. Applications to various random walks on groups, and the product replacement random algorithm, are also presented.

We present a survey of recent developments of the Beilinson–Lusztig–MacPherson approach in the study of quantum gl n , infinitesimal quantum gl n , quantum gl ∞ and their associated q-Schur algebras, little q-Schur algebras and infinite q-Schur algebras. We also use the relationship between quantum gl ∞ and infinite q-Schur algebras to discuss their representations.

In 1980s, Gross–Zagier [GZ86] established a formula that relates the Neron–Tate height of Heegner points on modular curves to the central derivative of certain L-functions associated to modular forms. Around the same time, Waldspurger proved a formula, relating toric periods of modular forms to the central value of certain L-functions. Gross put both of these formula in the framework of represe...

We prove a nonsymmetric analogue of a formula of Kato and Lusztig which describes the coefficients of the expansion of irreducible Weyl characters in terms of (degenerate) symmetric Macdonald polynomials as certain Kazhdan–Lusztig polynomials. We also establish precise polynomiality results for coefficients of symmetric and nonsymmetric Macdonald polynomials and a version of Demazure’s characte...

We provide a simple combinatorial proof of, and generalize, a theorem of Polo which asserts that for any polynomial P ∈ N[q] such that P (0) = 1 there exist two permutations u and v in a suitable symmetric group such that P is equal to the Kazhdan-Lusztig polynomial P v u .

In this note we explain how the computation of the spectrum of the lamplighter group from [GZ01] yields a counterexample to a strong version of the Atiyah conjectures about the range of L -Betti numbers of closed manifolds.

Let K be a field and let G be a multiplicative group. The group ring K[G] is an easily defined, rather attractive algebraic object. As the name implies, its study is a meeting place for two essentially different algebraic disciplines. Indeed, group ring results frequently require a blend of group theoretic and ring theoretic techniques. A natural, but surprisingly elusive, group ring problem co...

We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan–Lusztig polynomials. This consequence simple observation that one can use the solution Soergel's conjecture to make ambiguities involved in defining certain morphisms between Soergel bimodules characteristic zero (double leaves) disappear.