Matching subspaces in a field extension

Abstract

In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection f between two finite subsets A,B of G with the property, motivated by old questions on symmetric tensors, that the product af(a)does not belong to A for all a \in A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting. Given a skew field extension K \subset L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K-subspaces A,B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra.