Representations of symmetric groups on the homology of dual matroids of complete graphs

This thesis investigates the representations of the symmetric group on the homology of the dual matroid of a complete graph. These representations arise as follows: with each graph we can associate a matroid, by taking the set of edges of the graph as ground set and the edge sets of simple cycles as the circuits of the matroid. We focus on the dual of the matroid of the complete graph. We calculate the homology of the simplicial complex L associated with this matroid. Permuting the vertices of the complete graph induces a permutation on the edge set which is a vertex map of the simplicial complex. This vertex map sends independents to independents, thus inducing a simplicial map from the polytope of L to itself, hence on the homology spaces of L. This defines a representation of the symmetric group on the homology Hi(L,C). We show that the above representation is induced from a primitive representation of the cyclic subgroup of order n.