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 φ between two finite subsets A, B of G with the property, motivated by old questions on symmetric tensors, that a φ ( a ) ∉ A for all a ∈ 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 ⊂ 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.