Canonical reduced expression in affine Coxeter groups of type $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{D}_n$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908625356718080 |
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| author | Harbat, Sadek Al |
| author_facet | Harbat, Sadek Al |
| contents | We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde A$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower regardless of the ground ring. We give a similar canonical reduced expression for the elements of $W(\tilde{B}_n)$ and $W(\tilde{D}_n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical reduced expression in affine Coxeter groups of type $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{D}_n$ Harbat, Sadek Al Representation Theory Group Theory Primary 20F55, Secondary 05E16, 20C08 We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde A$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower regardless of the ground ring. We give a similar canonical reduced expression for the elements of $W(\tilde{B}_n)$ and $W(\tilde{D}_n)$. |
| title | Canonical reduced expression in affine Coxeter groups of type $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{D}_n$ |
| topic | Representation Theory Group Theory Primary 20F55, Secondary 05E16, 20C08 |
| url | https://arxiv.org/abs/2511.01702 |