An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups ...

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $picolon Rto R/J(R)$ be‎ the canonical map‎. ‎Then $pi$ induces an order preserving group homomorphism‎ ‎$K_0picolon K_0(R)to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0pi).$ We give a condition ...

If a polycyclic-by-finite rank-K group Γ admits a faithful affine representation making it acting on RK properly discontinuously and with compact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent group...

The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over Q. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is 3/2transitive and primitive, and that it is either almost...

The degenerate affine Hecke algebra (dAHA) of any finite Coxeter group was defined by Drinfeld and Lusztig([Dri],[Lus]). It is generated by the group algebra of the Coxeter group and by the commuting generators yi with some relations. In [AS], the authors give a Lie-theoretic construction of representations of the dAHA of type An−1. They construct a functor from the BGG category of slN to the c...

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given–three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index...

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G1;S1) and Cay(G2;S2) are connected Cayley graphs of finite valency on two nilpot...