2012/08/14 by Shalom Eliahou, Eliahou, Shalom, Cedric Lecouvey +1 · 3 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1208.2792
The present version corrects a slight gap in the statement of Theorem 2.6 of the published version of this paper [Journal of Algebra 324 (2010) 3420-3430]
arxiv created 2012/08/14 · arxiv updated 2012/08/15
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 ∈ 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.