Applying a classical theorem of Smith, we show that the poset property of being Gorenstein∗ over Z2 is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. ...

The Eulerian polynomial of a finite Coxeter system (W, S) of rank n records, for each k ∈ {1, . . . , n}, the number of elements w ∈ W with an ascent set {s ∈ S | l(ws) > l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the symmetric group on n + 1 letters. Victor Reiner gave a formula for...

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this condition for Coxeter groups in certain classes, possibly of infinite ranks. We also determine the finite irreducible components of another Coxeter group, t...

A Reidemeister-Schreier program which yields a presentation of a subgroup H of finite index in a finitely presented group G is described in [3J. The program has two stages; first, Schreier generators and Reidemeister relators for H are computed, then the resulting presentation is simplified by eliminating redundant generators and by using a substring searching technique. The Tietze transformati...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization o...

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many author...

We present a systematic study of the cosmological billiard structures of Einstein-p-form systems in which all spatial directions are compactified on a manifold of nontrivial topology. This is achieved for all maximally oxidised theories associated with split real forms, for all possible compactifications as defined by the de Rham cohomology of the internal manifold. In each case, we study the C...