We prove that the Kazhdan-Lusztig polynomials are combinatorial invariants for intervals up to length 8 in Coxeter groups of type A and up to length 6 in Coxeter groups of type B and D. As a consequence of our methods, we also obtain a complete classification, up to isomorphism, of Bruhat intervals of length 7 in type A and of length 5 in types B and D, which are not lattices. Résumé. On montre...

If ∆ is a polytope in real affine space, each edge of ∆ determines a reflection in the perpendicular bisector of the edge. The exchange group W (∆) is the group generated by these reflections, and ∆ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The GelfandSerganova Theorem and the st...

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient H of H and show that it has a basis parametrized by a certain subset Wc of the Coxeter group w. Specifically, Wc consists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommu...

** Associate Professor, University of Maryland; National Fellow, Hoover Institution, Stanford University. We are grateful to Ron Gilson, Joe Grundfest, Alan Schwartz, and Eric Talley for helpful comments. Todd Cleary provided excellent research assistance. The support of the John M. Olin Program in Law and Economics at Stanford Law School and The Roberts Center for Law, Business, and Corporate ...

We prove that an infinite irreducible Coxeter group cannot be a non-trivial direct product. Let W be a Coxeter group, and write W = W1 × · · · ×Wp ×Wp+1, where W1, . . . ,Wp are infinite irreducible Coxeter groups, and Wp+1 is a finite one. As an application of the main result, we obtain that W1, . . . ,Wp are unique and Wp+1 is unique up to isomorphism. That is, if W = W̃1 × · · · × W̃q × W̃q+1 i...

We give a new proof of Brink’s theorem that the nonreflection part of a reflection centralizer in a Coxeter group is free, and make several refinements. In particular we give an explicit finite set of generators for the centralizer and a method for computing the Coxeter diagram for its reflection part. In many cases, our method allows one to compute centralizers quickly in one’s head. We also d...

Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shor...

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Introduction The central object of this paper is the growth series of a Coxeter group. Many geometric features of such a group (or any group i...

The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors ...