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arXiv · 2507.14033

Shape and class of Bruhat Intervals

Gaston Burrull·Nicolas Libedinsky·Rodrigo Villegas
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Abstract

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.

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Gaston Burrull, Nicolas Libedinsky, Rodrigo Villegas. 2025-07-18. Shape and class of Bruhat Intervals. https://arxiv.org/abs/2507.14033

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