Factorization Combinatorics in Complex Reflection Groups
Open AccessLet $G$ be a complex reflection group. If $f=(t_1, \ldots, t_k)$ is a tuple of elements in $G$ such that $g=t_1\cdots t_k$, then $f$ is a \emph{factorization} of $g$. In the first part of this thesis, we study factorizations of an arbitrary element in $G$ whose factors are reflections in $G$. Specifically, we focus on how factorizations of a fixed element relate to one another via an operation called the \emph{Hurwitz action}. When $G$ is a real reflection group, Baumeister, Gobet, Roberts and Wegener classified the elements in $G$ that are \emph{Hurwitz-transitive}, meaning that their minimum-length factorizations form a single orbit under the Hurwitz action. In the complex setting, we characterize the Hurwitz-transitive elements when $G=G(m,p,n)$. Further, we give a counting formula that computes the number of orbits that the minimum-length factorizations of an arbitrary element in $G=G(m,p,n)$ form under the Hurwitz action. In the second part, we restrict the factors $t_i$ in a factorization $f$ to belong to a minimum generating set of reflections in $G$. In this context, $f$ is also referred to as a \emph{reduced expression} of $g$. We study how factorizations of a fixed element relate to one another via only commutation relations. An element is \emph{fully commutative} if its reduced expressions are all equivalent via commutation relations. Stembridge gave counting formulas and characterizations of fully commutative elements in $\type{B}_n=G(2,1,n)$ and $\type{D}_n=G(2,2,n)$. When $G=G(m,1,n)$, we give a counting formula for fully commutative elements and a characterization of these elements by pattern avoidance. When $G=G(m,m,n)$, we introduce several minimum generating sets for $G$ and show that pattern avoidance fails to characterize fully commutative elements. In the third part, we relax the condition on the factors $t_i$ being reflections in $G$. In the symmetric group $\Symm_n$, a permutation can be written uniquely (up to the order of factors) as a factorization of cycles. Gobet showed that the cycle decomposition of permutations in $\Symm_n$ can be generalized to Coxeter groups. In particular, he studied a class of elements in Coxeter groups that have unique (up to the order of factors) \emph{generalized cycle decompositions}. We initiate the study of elements in $G(m,p,n)$ whose generalized cycle decompositions are unique.
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