Lagmental Vicfred

The Group Algebra Converts Representations into Modules by Vicfred

tags: mathematics, representation theory, characters, group algebras

Linear representations of a group over k are the same objects as left modules over the group algebra k of G. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The data

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.

$$ k[G]=\left\{\sum_{g\in G}a_gg:a_g\in k\right\} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ k[G]\text{-Mod}\simeq\operatorname{Rep}_k(G) $$

Derivation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \left(\sum_ga_gg\right)\!\left(\sum_hb_hh\right)=\sum_{x\in G}\left(\sum_{gh=x}a_gb_h\right)x $$

Invariant content

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad k[G]=\left\{\sum_{g\in G}a_gg:a_g\in k\right\},\\[5pt] \mathsf{C}\;&:\quad k[G]\text{-Mod}\simeq\operatorname{Rep}_k(G). \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] k[G]\text{-Mod}\simeq\operatorname{Rep}_k(G) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Mon 14 November 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.