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The Regular Representation Contains Every Irreducible by Vicfred

Last updated: Fri 03 October 2025

tags: mathematics, representation theory, characters, group algebras

For a finite group over the complex numbers, each irreducible appears in the regular representation with multiplicity equal to its dimension. I want the notation, the mechanism, and the failure mode visible at the same time.

Start locally

A representation \(\rho:G\to\operatorname{GL}(V)\) replaces group elements by linear maps. Its character \(\chi_\rho(g)=\operatorname{tr}\rho(g)\) forgets bases while retaining remarkable decomposition data.

$$ \lambda:G\to\operatorname{GL}(\mathbf C[G]),\qquad\lambda(g)e_h=e_{gh} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \mathbf C[G]\cong\bigoplus_i(\dim V_i)V_i $$

Compute before generalising

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ |G|=\sum_i(\dim V_i)^2,\qquad\chi_{\mathrm{reg}}(g)=\begin{cases}|G|,&g=e,\\0,&g\ne e.\end{cases} $$

The global view

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \lambda:G\to\operatorname{GL}(\mathbf C[G]),\qquad\lambda(g)e_h=e_{gh},\\[5pt] \mathsf{C}\;&:\quad \mathbf C[G]\cong\bigoplus_i(\dim V_i)V_i. \end{aligned} $$

Edge conditions

Maschke's theorem needs \(\operatorname{char}k\nmid|G|\). In modular characteristic, invariant subspaces need not have invariant complements.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf C[G]\cong\bigoplus_i(\dim V_i)V_i \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sun 19 March 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.