The free $m$-cone of a matroid and its $\mathcal{G}$-invariant
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| Format: | Preprint |
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2021
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| _version_ | 1866910556156329984 |
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| author | Bonin, Joseph E. Long, Kevin |
| author_facet | Bonin, Joseph E. Long, Kevin |
| contents | For a matroid $M$, its configuration determines its $\mathcal{G}$-invariant. Few examples are known of pairs of matroids with the same $\mathcal{G}$-invariant but different configurations. In order to produce new examples, we introduce the free $m$-cone $Q_m(M)$ of a loopless matroid $M$, where $m$ is a positive integer. We show that the $\mathcal{G}$-invariant of $M$ determines the $\mathcal{G}$-invariant of $Q_m(M)$, and that the configuration of $Q_m(M)$ determines $M$; so if $M$ and $N$ are nonisomorphic and have the same $\mathcal{G}$-invariant, then $Q_m(M)$ and $Q_m(N)$ have the same $\mathcal{G}$-invariant but different configurations. We prove analogous results for several variants of the free $m$-cone. We also define a new matroid invariant of $M$, and show that it determines the Tutte polynomial of $Q_m(M)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_00046 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The free $m$-cone of a matroid and its $\mathcal{G}$-invariant Bonin, Joseph E. Long, Kevin Combinatorics 05B35 For a matroid $M$, its configuration determines its $\mathcal{G}$-invariant. Few examples are known of pairs of matroids with the same $\mathcal{G}$-invariant but different configurations. In order to produce new examples, we introduce the free $m$-cone $Q_m(M)$ of a loopless matroid $M$, where $m$ is a positive integer. We show that the $\mathcal{G}$-invariant of $M$ determines the $\mathcal{G}$-invariant of $Q_m(M)$, and that the configuration of $Q_m(M)$ determines $M$; so if $M$ and $N$ are nonisomorphic and have the same $\mathcal{G}$-invariant, then $Q_m(M)$ and $Q_m(N)$ have the same $\mathcal{G}$-invariant but different configurations. We prove analogous results for several variants of the free $m$-cone. We also define a new matroid invariant of $M$, and show that it determines the Tutte polynomial of $Q_m(M)$. |
| title | The free $m$-cone of a matroid and its $\mathcal{G}$-invariant |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2106.00046 |