Local Schur’s Lemma and Commutative Semifields

Abstract

A set of linear maps $\cal{R}\subset GL(V,K)$, V a finite vector space over a field K, is regular if to each $x,y\in V^*$ there corresponds a unique element $R\in\cal{R}$ such that R(x)=y. In this context, Schur’s lemma implies that $\overline{\cal{R}}=\cal{R}\cup \{ 0\}$ is a field if (and only if) it consists of pairwise commuting elements. We consider when $\cal{R}$ is locally commutative: at some μ ∈V*, AB(μ)=BA(μ) for all $A,B \in \cal{R}$, and $\cal{R}$ has been normalized to contain the identity. We show that such locally commutative $\cal{R}$ are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative $\cal{R}$ on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative $\cal{R}$. We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set $\cal{R}$ implies that the set of Lie products $[\cal{R},\cal{R}]$ consists entirely of singular maps, the converse is false.