The geometry of conjugation in affine Coxeter groups - Milićević, Elizabeth; Schwer, Petra; Thomas, Anne (International journal of algebra and computation, 2025)
We develop precise geometric descriptions of the conjugacy class [x] and coconjugation set C(x,x')={y∈W̄|yxy−1=x'} for all elements x,x' of any affine Coxeter group W̄. 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 ModW̄(w)=(w−I)R∨, where w is the finite part of x and R∨ is the coroot lattice. The set C(x,x') is then described by ModW̄(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 ModW̄(w) equals the dimension of Mov(w) and investigate type-by-type the surprisingly subtle structure of Mod W̄(w).