Counting Irreducible Representations of a Finite Abelian Group
Thomas Breuer, Prashun Kumar, Geetha Venkataraman
TL;DR
The paper solves the problem of counting irreducible representations of a finite abelian group $G$ over a finite field ${\mathbb F}_q$ of order $q$ with ${\rm char}({\mathbb F}_q)\nmid |G|$. It develops a Brauer-character based framework that assigns irreducible ${\mathbb F}_q$-representations to divisors $d$ of the exponent $e$ of $G$, with each such representation having degree ${\rm ord}(q\bmod d)$ and arising in orbits under the Galois action, yielding the counting formula $(1/n)\sum_{d\in D_n} |{\rm I}_d(G)|$ for the number of irreducibles of degree $n$. The counts factor over Sylow subgroups, giving $|{\rm I}_d(G)|=\prod_{r} |{\rm I}_{d_r}(G_r)|$, and the sizes ${|{\rm I}_{r^l}(G_r)|}$ are computable via kernels $N(G_r,l)$ and quotients $F(G_r,l)$, yielding degrees as ${\rm lcm}$s of ${\rm ord}(q\bmod r^{i_r})$. An explicit algorithm is provided to compute all irreducible degrees and their multiplicities from the prime-power decomposition of $|G|$, together with an example illustrating the method over particular finite fields. Overall, the work gives a concrete, structure-agnostic method to enumerate modular irreducibles of abelian groups and clarifies how degrees and multiplicities depend on the exponent and field data.
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.