The Whitney Algebra of a Matroid

The concept of matroid, with its companion concept of geometric lattice, was distilled by Hassler Whitney [19], Saunders Mac Lane [10] and Garrett Birkhoff [2] from the common properties of linear and algebraic dependence. The inverse problem, how to represent a given abstract matroid as the matroid of linear dependence of a specified set of vectors over some field (or as the matroid of algebraic dependence of a specified set of algebraic functions) has already prompted fifty years of intense effort by the leading researchers in the field: William Tutte, Dominic Welsh, Tom Brylawski, Neil White, Bernt Lindstrom, Peter Vamos, Joseph Kung, James Oxley, and Geoff Whittle, to name only a few. (A goodly portion of this work aimed to provide a proof or refutation of what is now, once again, after a hundred or so years, the 4-color theorem.) One way to attack this inverse problem, the representation problem for matroids, is first to study the ‘play of coordinates’ in vector representations. In a vector representation of a matroid M , each element of M is assigned a vector in such a way that dependent (resp., independent) subsets of M are assigned dependent (resp., independent) sets of vectors. The coefficients of such linear dependencies are computable as minors of the matrix of coordinates of the dependent sets of vectors; this is Cramer’s rule. For instance, if three points a, b, c are represented in R (that is, in real projective 3-space) by the dependent vectors forming the rows of the matrix C =  1 2 3 4 a 1 4 0 6 b −2 3 1 −5 c −4 17 3 −3 ,