Primitive elements in the matroid-minor Hopf algebra

We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis and coproduct given by splitting a matroid into a submatroid and complementary contraction in all possible ways. We introduce two new bases for C; the first of these is related to the distinguished basis by Möbius inversion over the rank-preserving weak order on matroids, the second by Möbius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is a basis for the subspace of primitive elements of C. Projecting C onto the matroid-minor Hopf algebra H, we obtain bases for the subspace of primitive elements of H.