UC Santa Barbara

Artin groups are generalizations of Emil Artin’s braid groups, and dual Artin groups are a variation defined using the Hurwitz action on reduced reflection factorizations of a fixed Coxeter element of the corresponding Coxeter group. There is a surjective group homomorphism from the Artin group to the corresponding dual Artin group. It is conjectured that the standard and dual Artin groups are always isomorphic. When the two groups are isomorphic, the properties of the dual Artin group can be used to tackle the word problem and K(π, 1) conjecture for the standard Artin group.This dissertation investigates the matrix representations and change of basis induced by the Hurwitz action on reduced reflection factorizations of Coxeter elements. When attention is restricted to powers of half-twists, the resulting matrix representations exhibit connections to Fibonacci and Chebyshev polynomials, and can be viewed geometrically as acting via rotations on certain planes in the standard geometric representation of the Coxeter system. Generalizing to powers of braids that twist a sequence of adjacent strands, the matrix representations exhibit connections to a generalization of the Fibonacci polynomials called polynomial wave sequences. This is a first step towards understanding all dual Artin groups in a uniform way.