Maximal nests of subspaces, the matrix Bruhat decomposition, and the marriage theorem - with an application to graph coloring

Abstract

Using the celebrated Marriage Theorem of P. Hall, we give an elementary combinato- rial proof of the theorem that asserts that given two maximal nests N1 and N2 in a finite dimensional vector space V , there is an ordered basis of V that generates N1 and a permutation of that ordered basis that generates N2. From this theorem one easily obtains the Matrix Bruhat Decomposition. A generalization to matroids is discussed, and an application to graph coloring is given.