Combinatorial Aspects of the Theory of Canonical Forms

A geometric model for a class of bipartite graphs is introduced, and a type of perfect matching, called an acyclic matching, is defined and through geometric reasoning shown to exist for a subset of the bipartite graphs discussed. These acyclic matchings imply a nonvanishing determinant for a class of weighted biadjacency matrices. This matching theory is applied to address a question raised by E. K. Wakeford in 1916, on the possible sets of monomials which can be removed from a generic homogeneous polynomial through linear changes in its variables. The notion of essential rank for the p-th graded piece of the exterior algebra is given a geometric interpretation. It is shown that essential rank gives information about the Pliicker embedding of the Grassmannian G(p, V) in projective space over AP(V). The Lottery problem is then discussed, and its relationship to the essential rank of AP(V) is explained. Thesis Supervisor: Gian-Carlo Rota Title: Professor of Applied Mathematics and Philosophy