Counting Irreducible Representations of a Finite Abelian Group

Thomas Breuer; Prashun Kumar; Geetha Venkataraman

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arXiv · 2504.12787

Counting Irreducible Representations of a Finite Abelian Group

Thomas Breuer·Prashun Kumar·Geetha Venkataraman

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Abstract

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension $n$ over the field of order $q$, up to equivalence, using Brauer characters. We also provide a formula for such $n$ using the prime decomposition of the exponent of $G$ and an algorithm to compute the irreducible degrees and their multiplicities.

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Thomas Breuer, Prashun Kumar, Geetha Venkataraman. 2025-04-17. Counting Irreducible Representations of a Finite Abelian Group. https://arxiv.org/abs/2504.12787

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