Tropical representations and valuated matroids
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910720832045056 |
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| author | Jun, Jaiung Mincheva, Kalina Tolliver, Jeffrey |
| author_facet | Jun, Jaiung Mincheva, Kalina Tolliver, Jeffrey |
| contents | We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a semiring: weakly free and quasi-free modules. We also investigate subgroups of $\text{GL}_n(K)$ for $K=\mathbb{T}$, $ \mathbb{R}_{\geq 0}$, and automorphisms of weakly free modules and tropical prevarieties defined by tropical linear equations. As an application of our results, we provide an intrinsic description of tropical subrepresentation via certain quasi-free modules, and prove that a tropical subrepresentation is equivalent to a valuated matroidal representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical representations and valuated matroids Jun, Jaiung Mincheva, Kalina Tolliver, Jeffrey Representation Theory Algebraic Geometry Combinatorics Rings and Algebras 12K10, 14T10, 05B35, 05E10 We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a semiring: weakly free and quasi-free modules. We also investigate subgroups of $\text{GL}_n(K)$ for $K=\mathbb{T}$, $ \mathbb{R}_{\geq 0}$, and automorphisms of weakly free modules and tropical prevarieties defined by tropical linear equations. As an application of our results, we provide an intrinsic description of tropical subrepresentation via certain quasi-free modules, and prove that a tropical subrepresentation is equivalent to a valuated matroidal representation. |
| title | Tropical representations and valuated matroids |
| topic | Representation Theory Algebraic Geometry Combinatorics Rings and Algebras 12K10, 14T10, 05B35, 05E10 |
| url | https://arxiv.org/abs/2411.19889 |