  A G-gain graph is a graph whose oriented edges are labeled invertibly from a
group G. Zaslavsky proposed two matroids of G-gain graphs, called frame
matroids and lift matroids, and investigated linear representations of them.
Each matroid has a canonical representation over a field F if G is isomorphic
to a subgroup of F^{\times} in the case of frame matroids or G is isomorphic to
an additive subgroup of 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.
  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
G. 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 copies of a frame matroid or a lift matroid, based on linear
representations of G on a d-dimensional vector space. We show that linear
matroids of the latter extension are indeed special cases of the first
extensions, 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 from recent research on 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.
