Shape and class of Bruhat Intervals
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911064012095488 |
|---|---|
| author | Burrull, Gaston Libedinsky, Nicolas Villegas, Rodrigo |
| author_facet | Burrull, Gaston Libedinsky, Nicolas Villegas, Rodrigo |
| contents | We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type $\widetilde{A}_2$ we show that each interval coincides with a generalized permutohedron minus a star-shaped polygon, and we prove a subtler version inside the dominant chamber of type $\widetilde{A}_n$. Motivated by this geometry, we conjecture that whenever two Bruhat intervals are isomorphic, there exists an isomorphism realized by a piecewise isometry. We prove this when both endpoints are dominant in $\widetilde{A}_2$ and obtain partial results in $\widetilde{A}_n$. In the course of proving these results, we made the surprising observation that much of the information contained in a Bruhat interval is already encoded in a tiny portion of it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape and class of Bruhat Intervals Burrull, Gaston Libedinsky, Nicolas Villegas, Rodrigo Combinatorics Representation Theory 05E10 (Primary), 20F55 (Secondary) We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type $\widetilde{A}_2$ we show that each interval coincides with a generalized permutohedron minus a star-shaped polygon, and we prove a subtler version inside the dominant chamber of type $\widetilde{A}_n$. Motivated by this geometry, we conjecture that whenever two Bruhat intervals are isomorphic, there exists an isomorphism realized by a piecewise isometry. We prove this when both endpoints are dominant in $\widetilde{A}_2$ and obtain partial results in $\widetilde{A}_n$. In the course of proving these results, we made the surprising observation that much of the information contained in a Bruhat interval is already encoded in a tiny portion of it. |
| title | Shape and class of Bruhat Intervals |
| topic | Combinatorics Representation Theory 05E10 (Primary), 20F55 (Secondary) |
| url | https://arxiv.org/abs/2507.14033 |