This paper gives some new examples in the 1-cohomology theory of finite groups of Lie type, obtained from both computer calculations and the use of several theoretical results. In particular, the paper gives the first known examples of 1-cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group. The computer calculations were made originally whi...

We use the polynomial ring C[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Snmodules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Claus...

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. 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 [16] has given a framework for computing the Kazhdan-Lusztig polynomials which generally invo...

For more than one hundred years, Schubert decomposition of complete and partial flag manifolds and Schubert calculus have been an intertwining point of algebra, geometry, representation theory, and combinatorics. In this paper, we are concerned with the further refinement of Schubert decomposition theory. Our far and ambitious goal is to describe the topology of intersection of two arbitrary Sc...

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.