Exponential matrices

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 a study of exponential matrices in positive characteristic. 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.