NDLI: Local Schur’s Lemma and Commutative Semifields

Content Provider	Springer Nature Link
Author	Jha, Vikram
Copyright Year	2005
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.
Starting Page	203
Ending Page	216
Page Count	14
File Format	PDF
ISSN	09251022
Journal	Designs, Codes and Cryptography
Volume Number	36
Issue Number	2
e-ISSN	15737586
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2005-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	nets Schur’s lemma semifields translation planes Data Structures, Cryptology and Information Theory Data Encryption Coding and Information Theory Discrete Mathematics in Computer Science Information and Communication, Circuits Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science Applications

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