Conjugacy of Coxeter elements

For a Coxeter group (W,S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter elements are rotation equivalent if their words can be transformed into each other through a sequence of rotations and legal commutations. We prove that Coxeter elements are conjugate if and only if they are rotation equivalent. This was known for some special cases but not for Coxeter groups in general

Reduced words in affine Coxeter groups

AbstractLet r(w) denote the number of reduced words for an element w in a Coxeter group W. Stanley proved a formula for r(w) when W is the symmetric group An, and he suggested looking at r(w) for the affine group An. We prove that for any affine Coxeter group Xn there is a finite number of types of elements in Xn, such that to every element w can be associated (1) a type t, (2) an element v in the finite group Xn, and (3) an n-tuple (m1,m2,…,mn) of integers mi ⩾ 0. Then r(w) = rvt(m1,…,mn), and for every rvt and for large enough mi, a homogeneous linear n-dimensional recurrence holds. For An, this takes a nice combinatorial form. We also discuss a canonical reduced word for w associated to its n-tuple

Strong Convergence and a Game of Numbers

AbstractS. Mozes investigated a certain solitary game played on a weighted graph. Numbers are placed on the nodes of the graph, and a move consists of changing the sign of a negative number and changing the numbers on the neighboring nodes according to the weights on the edges. Mozes proved that the game has a strong convergence property when the edges have certain positive integer weights. However, his approach would give no information in the case of other weights. In this paper we first prove that strong convergence is equivalent to the fact that the game has as a certain ‘polygon property’. We can then, in a rather elementary way, characterize the assignments of weights that imply the polygon property, and hence strong convergence. Finally, we make a natural generalization of the game, where we also have weights on the nodes. The conditions for strong convergence generalize nicely to this game