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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.28634 |
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| _version_ | 1866910222507835392 |
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| author | Li, Zijun |
| author_facet | Li, Zijun |
| contents | We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N].
We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complexes. Under this correspondence, the comultiplication structure of C[N] matches the intrinsic comultiplication structure of folded galleries given by projections to sub-Coxeter complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A path model for MV polytopes in type A_n Li, Zijun Representation Theory 17B10 (Primary) 20G05, 05E14, 13F60 (Secondary) We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N]. We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complexes. Under this correspondence, the comultiplication structure of C[N] matches the intrinsic comultiplication structure of folded galleries given by projections to sub-Coxeter complexes. |
| title | A path model for MV polytopes in type A_n |
| topic | Representation Theory 17B10 (Primary) 20G05, 05E14, 13F60 (Secondary) |
| url | https://arxiv.org/abs/2603.28634 |