The geometry of conjugation in affine Coxeter groups
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 [...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
2025
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| In: |
International journal of algebra and computation
Year: 2025, Volume: 35, Issue: 03, Pages: 403-465 |
| ISSN: | 0218-1967 |
| DOI: | 10.1142/S0218196725500109 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1142/S0218196725500109 Verlag, lizenzpflichtig, Volltext: https://www.worldscientific.com/doi/10.1142/S0218196725500109 
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| Author Notes: | Elizabeth Milićević, Petra Schwer, and Anne Thomas |
| Summary: | 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). |
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| Item Description: | Gesehen am 02.10.2025 |
| Physical Description: | Online Resource |
| ISSN: | 0218-1967 |
| DOI: | 10.1142/S0218196725500109 |