The geometry of conjugation in affine Coxeter groups

Abstract

We develop new and precise geometric descriptions of the conjugacy class $[x]$ and coconjugation set $C(x,x^{\prime })=\{y\in \overline{W}\mid yx{y}^{-1}=x^{\prime }\}$ for all elements $x,x^{\prime }$ of any affine Coxeter group $\overline{W}$. The centralizer of $x$ in $\overline{W}$ is the special case $C(x,x)$. The key structure in our description of the conjugacy class $[x]$ is the mod-set ${Mod}_{\overline{W}}(w)=(w-I)R^{\vee }$, where~$w$ is the finite part of $x$ and $R^{\vee }$ is the coroot lattice. The coconjugation set $C(x,x^{\prime })$ is then described by ${Mod}_{\overline{W}}(w^{\prime })$ together with the fix-set of $w^{\prime }$, where $w^{\prime }$ is the finite part of $x^{\prime }$. For any element $w$ of the associated finite Weyl group $W$, the mod-set of $w$ is contained in the classical move-set $Mov(w)=\mathrm{Im}(w-I)$. 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.

116 pages, 6 figures best viewed in color; v3: minor revisions, shorter version to appear in the International Journal of Algebra and Computation (IJAC)