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arXiv · 2209.00066

Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

Theo Douvropoulos·Joel Brewster Lewis·Alejandro H. Morales
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Abstract

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of reduced reflection factorizations of $g$ and $\operatorname{RGS}(W,g)$ of the relative generating sets of $g$. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size $\#\operatorname{Red}_W(g)$ with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-$0$ Hurwitz numbers.

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Theo Douvropoulos, Joel Brewster Lewis, Alejandro H. Morales. 2022-08-31. Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements. https://doi.org/10.1016/j.jalgebra.2023.11.015

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