## University of Sydney Algebra Seminar

# Xiang Fu (University of Sydney)

## Friday 23rd November, 12:05-12:55pm, Carslaw 173

### The dominance partial ordering in Coxeter groups

In this talk we present a few new findings concerning a partial ordering, called *dominance*, defined on the root system of an arbitrary Coxeter group. Dominance was first introduced by Brink and Howlett in their proof that finitely generated Coxeter groups are *automatic*, and it was later utilized as a tool in general combinatorial and algebraic investigations of Coxeter groups. However, in the literature only the set of roots being minimal with respect to dominance, called *elementary roots*, were investigated, and in this talk we will make attempts to describe the behaviour of dominance beyond these elementary roots. Time permitting, we will also outline how dominance in an arbitrary Coxeter group can be reasonably characterized by its behaviour in dihedral reflection subgroups of that Coxeter group. These results were taken from our recent papers in J. Alg and PJM.