The geometry of conjugation in affine Coxeter groups
Copyright policy )The geometry of conjugation in affine Coxeter groups
We develop precise geometric descriptions of the conjugacy class [Formula: see text] and coconjugation set [Formula: see text] for all elements [Formula: see text] of any affine Coxeter group [Formula: see text]. The centralizer of x in [Formula: see text] is the special case [Formula: see text]. The key structure in our description of the conjugacy class [Formula: see text] is the mod-set [Formula: see text], where w is the finite part of x and [Formula: see text] is the coroot lattice. The set [Formula: see text] is then described by [Formula: see text] together with the fix-set of [Formula: see text], where [Formula: see text] is the finite part of [Formula: see text]. For any element w of the associated finite Weyl group W, the mod-set of w is contained in the classical move-set [Formula: see text]. We prove that the rank of [Formula: see text] equals the dimension of [Formula: see text] and investigate type-by-type the surprisingly subtle structure of [Formula: see text].
- Heidelberg University Germany
- Haverford College United States
- University of Sydney Australia
affine Coxeter group, Other geometric groups, including crystallographic groups, Group Theory (math.GR), centralizer, Combinatorial aspects of groups and algebras, crystallographic group, Reflection and Coxeter groups (group-theoretic aspects), FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Representation Theory (math.RT), Mathematics - Group Theory, reflection group, Mathematics - Representation Theory, Conjugacy classes for groups, conjugation
affine Coxeter group, Other geometric groups, including crystallographic groups, Group Theory (math.GR), centralizer, Combinatorial aspects of groups and algebras, crystallographic group, Reflection and Coxeter groups (group-theoretic aspects), FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Representation Theory (math.RT), Mathematics - Group Theory, reflection group, Mathematics - Representation Theory, Conjugacy classes for groups, conjugation
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