The geometry of conjugation in affine Coxeter groups
Elizabeth Milićević, Petra Schwer, Anne Thomas
TL;DR
This work develops a precise geometric framework for conjugacy and coconjugation in affine Coxeter groups ${\overline{W}} = T \rtimes W$ by introducing mod-sets ${\operatorname{Mod}}_{\overline{W}}(w)$ and relating them to move-sets ${\operatorname{Mov}}(w)$. A central result shows that the rank of ${\operatorname{Mod}}_{\overline{W}}(w)$ equals the move-set dimension, equal to the finite Weyl-group reflection-length, and that ${\operatorname{Mod}}_{\overline{W}}(w)$ sits as a finite-index submodule of ${\operatorname{Mov}}(w) \cap R^{\vee}$; this provides a coarse geometric picture of conjugacy classes via translations and $W$-conjugation. The authors supply explicit, type-by-type descriptions of ${\operatorname{Mod}}_{\overline{W}}(w)$ for types ${A_n}$, ${C_n}$, ${B_n}$, and ${D_n}$, including $\,\mathbb{Z}$-bases, Smith normal forms, and the resulting isomorphism types of $R^{\vee}/\textsc{Mod}(w)$; they also characterize when ${\operatorname{Mod}}(w) = {\operatorname{Mov}}(w) \cap R^{\vee}$ (fill move-sets). Extending to split crystallographic subgroups (MST4), they provide a unified geometric description of conjugacy classes and coconjugation sets across affine and crystallographic settings. These results yield a constructive toolkit for describing conjugacy classes and centralizers in affine Weyl groups with potential applications to Hecke algebras, Deligne–Lusztig theory, and related geometric representation theories.
Abstract
We develop new and precise geometric descriptions of the conjugacy class $[x]$ and coconjugation set $\operatorname{C}(x,x') = \{ y \in \overline{W} \mid yxy^{-1} = x' \}$ for all elements $x,x'$ of any affine Coxeter group $\overline{W}$. The centralizer of $x$ in $\overline{W}$ is the special case $\operatorname{C}(x,x)$. The key structure in our description of the conjugacy class $[x]$ is the mod-set ${Mod}_{\overline{W}}(w) = (w-\operatorname{I})R^\vee$, where~$w$ is the finite part of $x$ and $R^\vee$ is the coroot lattice. The coconjugation set $\operatorname{C}(x,x')$ is then described by ${Mod}_{\overline{W}}(w')$ together with the fix-set of $w'$, where $w'$ is the finite part of $x'$. For any element $w$ of the associated finite Weyl group $W$, the mod-set of $w$ is contained in the classical move-set ${Mov}(w) = \operatorname{Im}(w - \operatorname{I})$. We prove that the rank of ${Mod}_{\overline{W}}(w)$ equals the dimension of ${Mov}(w)$, and then further investigate type-by-type the surprisingly subtle structure of the $\mathbb{Z}$-module ${Mod}_\overline{W}(w)$. As corollaries, we determine exactly when ${Mod}_{\overline{W}}(w) = {Mov}(w) \cap R^\vee$, in which case our closed-form descriptions of conjugacy classes and coconjugation sets are as simple as possible.