Polymatroids are to finite groups as matroids are to finite fields
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arXiv
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| Natura: | Preprint |
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2024
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| author | Swartz, Ed Wentworth-Nice, Prairie Xue, Alexander |
| author_facet | Swartz, Ed Wentworth-Nice, Prairie Xue, Alexander |
| contents | Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} Γ_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $Γ_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row space equals $\mathcal{H}.$ In general, there are many ways in which the relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams identities when $\mathcal{H}$ is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by $\mathcal{G}$ and $\mathcal{H}.$ We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polymatroids are to finite groups as matroids are to finite fields Swartz, Ed Wentworth-Nice, Prairie Xue, Alexander Combinatorics Group Theory 05B35 (Primary) 94B60, 05C15 (Secondary) Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} Γ_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $Γ_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row space equals $\mathcal{H}.$ In general, there are many ways in which the relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams identities when $\mathcal{H}$ is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by $\mathcal{G}$ and $\mathcal{H}.$ We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs. |
| title | Polymatroids are to finite groups as matroids are to finite fields |
| topic | Combinatorics Group Theory 05B35 (Primary) 94B60, 05C15 (Secondary) |
| url | https://arxiv.org/abs/2402.17582 |