Matroids from gain graphs over quotient groups
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912928450478080 |
|---|---|
| author | Walsh, Zach |
| author_facet | Walsh, Zach |
| contents | We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $Γ$ with Frobenius kernel $Γ_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $Γ/Γ_1$.
While the hypothesis that $Γ$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $Γ$ is any finite group with a nontrivial proper normal subgroup $Γ_1$ and there is a construction that takes in a complete $Γ$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(Γ/Γ_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $Γ$-balanced, then $Γ$ is a Frobenius group with Frobenius kernel $Γ_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23066 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matroids from gain graphs over quotient groups Walsh, Zach Combinatorics Group Theory 05B35, 20B05 We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $Γ$ with Frobenius kernel $Γ_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $Γ/Γ_1$. While the hypothesis that $Γ$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $Γ$ is any finite group with a nontrivial proper normal subgroup $Γ_1$ and there is a construction that takes in a complete $Γ$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(Γ/Γ_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $Γ$-balanced, then $Γ$ is a Frobenius group with Frobenius kernel $Γ_1$. |
| title | Matroids from gain graphs over quotient groups |
| topic | Combinatorics Group Theory 05B35, 20B05 |
| url | https://arxiv.org/abs/2602.23066 |