2025/07/18 by Gaston Burrull, Burrull, Gaston, Nicolás Libedinsky +3
Mathematics · #05E10 (Primary) #20F55 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2507.14033
openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type \widetildeA2 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 \widetildeAn. 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 \widetildeA2 and obtain partial results in \widetildeAn. 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.