Matroids of gain graphs in applied discrete geometry

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A Γ \Gamma -gain graph is a graph whose oriented edges are labeled invertibly from a group Γ \Gamma . Zaslavsky proposed two matroids associated with Γ \Gamma -gain graphs, called frame matroids and lift matroids, and investigated linear representations of them. Each matroid has a canonical representation over a field F \mathbb {F} if Γ \Gamma is isomorphic to a subgroup of F × \mathbb {F}^{\times } in the case of frame matroids or Γ \Gamma is isomorphic to an additive subgroup of F \mathbb {F} in the case of lift matroids. The canonical representation of the frame matroid of a complete graph is also known as a Dowling geometry, as it was first introduced by Dowling for finite groups Γ \Gamma . In this paper, we extend these matroids in two ways. The first one is extending the rank function of each matroid, based on submodular functions over Γ \Gamma . The resulting rank function generalizes that of the union of frame matroids or lift matroids. Another one is extending the canonical linear representation of the union of d d copies of a frame matroid or a lift matroid, based on linear representations of Γ \Gamma on a d d -dimensional vector space. We show that linear matroids of the latter extension are indeed special cases of the first extension, as in the relation between Dowling geometries and frame matroids. We also discuss an attempt to unify the extension of frame matroids and that of lift matroids. This work is motivated by recent results in the combinatorial rigidity of symmetric graphs. As special cases, we give several new results on this topic, including combinatorial characterizations of the symmetry-forced rigidity of generic body-bar frameworks with point group symmetries or crystallographic symmetries and the symmetric parallel redrawability of generic bar-joint frameworks with point group symmetries or crystallographic symmetries.