Exponential matrices

TL;DR

This paper introduces exponential matrices with polynomial and exponential properties, and classifies them in positive characteristic, leading to classifications of certain modular group representations.

Contribution

It defines exponential matrices and develops their classification in positive characteristic, including specific cases for Heisenberg groups and 4x4 matrices, advancing modular representation theory.

Findings

Classified exponential matrices of Heisenberg groups in positive characteristic.

Classified 4x4 exponential matrices in positive characteristic.

Derived classifications of modular representations of elementary abelian p-groups.

Abstract

In this article, we introduce a notion of an exponential matrix, which is a polynomial matrix with exponential properties, and a notion of an equivalence relation of two exponential matrices, and then we initiate to study classifying exponential matrices in positive characteristic, up to equivalence. We classify exponential matrices of Heisenberg groups in positive characteristic, up to equivalence. We also classify exponential matrices of size four-by-four in positive characteristic, up to equivalence. From these classifications, we obtain a classification of modular representations of elementary abelian $p$-groups into Heisenberg groups, up to equivalence, and a classification of four-dimensional modular representations of elementary abelian $p$-groups, up to equivalence.