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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.06830 |
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| _version_ | 1866912708042948608 |
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| author | Imamura, Koji Kawabuchi, Shinya Shiromoto, Keisuke |
| author_facet | Imamura, Koji Kawabuchi, Shinya Shiromoto, Keisuke |
| contents | In this paper we introduce a $q$-analogue of the single-element extensions of matroids for $q$-matroids, which we call one-dimensional extensions. To enumerate such extensions, we define a $q$-analogue of modular cuts and define a certain function which we call a modular cut selector. It assigns each newly appearing one-dimensional subspace to a modular cut. By using these notion, we prove the one-to-one correspondence between the one-dimensional extensions and the modular cut selectors. Furthermore, we define the canonnical representatives of the isomorphic class of the $q$-matroids, which enable us to enumerate non-isomorphic $q$-matroids without the paiwise isomorphism testing. As an application, we develop a classification algorithm for $q$-matroids, and classify all the $q$-matroids on ground spaces over $\mathbb{F}_2$ and $\mathbb{F}_3$ of dimension $4$ and $5$ respectively. We also determine some $5$-dimensional $q$-matroids related to the $q$-Fano plane, which is the $q$-analogue of the Fano plane, over $\mathbb{F}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the one-dimensional extensions of $q$-matroids Imamura, Koji Kawabuchi, Shinya Shiromoto, Keisuke Combinatorics In this paper we introduce a $q$-analogue of the single-element extensions of matroids for $q$-matroids, which we call one-dimensional extensions. To enumerate such extensions, we define a $q$-analogue of modular cuts and define a certain function which we call a modular cut selector. It assigns each newly appearing one-dimensional subspace to a modular cut. By using these notion, we prove the one-to-one correspondence between the one-dimensional extensions and the modular cut selectors. Furthermore, we define the canonnical representatives of the isomorphic class of the $q$-matroids, which enable us to enumerate non-isomorphic $q$-matroids without the paiwise isomorphism testing. As an application, we develop a classification algorithm for $q$-matroids, and classify all the $q$-matroids on ground spaces over $\mathbb{F}_2$ and $\mathbb{F}_3$ of dimension $4$ and $5$ respectively. We also determine some $5$-dimensional $q$-matroids related to the $q$-Fano plane, which is the $q$-analogue of the Fano plane, over $\mathbb{F}_2$. |
| title | On the one-dimensional extensions of $q$-matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.06830 |