Lagmental Vicfred

Maschke Averaging Produces an Invariant Projection by Vicfred

tags: mathematics, representation theory, characters, group algebras

When the group order is invertible in the field, every finite-dimensional representation is completely reducible. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

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.

$$ P:V\to W,\qquad P|_W=1_W $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \overline P=\frac1{|G|}\sum_{g\in G}\rho(g)P\rho(g)^{-1} $$

Derivation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \begin{aligned}\overline P\,\rho(h)&=\frac1{|G|}\sum_g\rho(g)P\rho(g^{-1}h)\\&=\rho(h)\overline P,\qquad \overline P|_W=1_W.\end{aligned} $$

Invariant content

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad P:V\to W,\qquad P|_W=1_W,\\[5pt] \mathsf{C}\;&:\quad \overline P=\frac1{|G|}\sum_{g\in G}\rho(g)P\rho(g)^{-1}. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \overline P=\frac1{|G|}\sum_{g\in G}\rho(g)P\rho(g)^{-1} \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Thu 02 November 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.