The geometry of conjugation in affine Coxeter groups
| Autor: | Milićević, Elizabeth, Schwer, Petra, Thomas, Anne |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 116 pages, 6 figures best viewed in color; v2: updated reference to arXiv:2407.08078v2 |
| Databáze: | arXiv |
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