2024/07/10 by Elizabeth Milićević, Petra Schwer, Milićević, Elizabeth +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2407.08080
openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We develop new and precise geometric descriptions of the conjugacy class [x] and coconjugation set C(x,x') = \ y ∈ W | yxy-1 = x' \ for all elements x,x' of any affine Coxeter group W. The centralizer of x in W is the special case C(x,x). The key structure in our description of the conjugacy class [x] is the mod-set Mod_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') is then described by Mod_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) = Im(w - I). We prove that the rank of Mod_W(w) equals the dimension of Mov(w), and then further investigate type-by-type the surprisingly subtle structure of the ℤ-module ModW(w). As corollaries, we determine exactly when Mod_W(w) = Mov(w) ∩ R^\vee, in which case our closed-form descriptions of conjugacy classes and coconjugation sets are as simple as possible.