Group actions on semimatroids

Emanuele Delucchi, Sonja Riedel

Published 2015-07-24Version 1

We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable Tutte polynomial and a poset which, in the realizable case, coincides with the poset of connected components of intersections of the associated toric arrangement. In this structural framework we recover and strongly generalize many enumerative results about arithmetic matroids, arithmetic Tutte polynomials and toric arrangements by finding new combinatorial interpretations beyond the realizable case. In particular, we thus find the first class of natural examples of nonrealizable arithmetic matroids. Moreover, if the group is abelian, then the action gives rise to a family of Z-modules which, under appropriate conditions, defines a matroid over Z. As a stepping stone to our results we also prove an extension of the cryptomorphism between semimatroids and geometric semilattices to the infinite case.