Toward a Classification of the Supercharacter Theories of $C_p\times C_p$

TL;DR

This paper classifies supercharacter theories of elementary abelian p-groups of order p^2, showing they mainly originate from automorphisms, and provides computational results for small primes to support conjectures for general primes.

Contribution

It characterizes supercharacter theories of elementary abelian p-groups of order p^2, linking them to automorphisms and specific product constructions, and offers computational evidence for broader conjectures.

Findings

Supercharacter theories from automorphisms can be characterized.

Supercharacter theories with certain properties arise from specific product constructions.

Computations for small primes support conjectures for all primes.

Abstract

In this paper, we study the superscharacter theories of elementary abelian $p$-groups of order $p^2$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^2$ that has a nonidentity superclass of size $1$ or a nonprincipal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2, 3, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.