Kleshchev multipartitions, affine Mirković-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914619666202624 |
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| author | Allen, Samantha Isaac, Jack Moscariello, Corinne Muth, Robert Uwase, Bella Deborah Walton, Lucas |
| author_facet | Allen, Samantha Isaac, Jack Moscariello, Corinne Muth, Robert Uwase, Bella Deborah Walton, Lucas |
| contents | We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirković--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_00421 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kleshchev multipartitions, affine Mirković-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$ Allen, Samantha Isaac, Jack Moscariello, Corinne Muth, Robert Uwase, Bella Deborah Walton, Lucas Representation Theory Combinatorics 17B37, 05E10, 20C08, 16G99, 17B67, 05E05, 20C30 We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirković--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction. |
| title | Kleshchev multipartitions, affine Mirković-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$ |
| topic | Representation Theory Combinatorics 17B37, 05E10, 20C08, 16G99, 17B67, 05E05, 20C30 |
| url | https://arxiv.org/abs/2606.00421 |