Local Schur’s Lemma and Commutative Semifields

Abstract

A set of linear maps 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 R such that R(x)=y. In this context, Schur's lemma implies that R =R \cup {l} is a field if (and only if) it consists of pairwise commuting elements. We consider when R is locally commutative: at some µ e V*, AB(µ)=BA(µ) for all A,B \in \ R, and R has been normalized to contain the identity. We show that such locally commutative 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 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 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 R implies that the set of Lie products [R, R] consists entirely of singular maps, the converse is false.