Lagmental Vicfred

Schur's Lemma Makes Irreducible Endomorphisms Scalar by Vicfred

Last updated: Thu 26 August 2021

tags: mathematics, representation theory, characters, group algebras

An intertwiner between irreducible representations is either zero or an isomorphism, and over an algebraically closed field endomorphisms are scalar. I want the notation, the mechanism, and the failure mode visible at the same time.

Definitions first

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.

$$ T:V\to W,\qquad T\rho_V(g)=\rho_W(g)T $$

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 \(\mathsf C\).

$$ V,W\ \text{irreducible}\Longrightarrow T=0\ \text{or}\ T\ \text{is an isomorphism} $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \ker T,\operatorname{im}T\ \text{invariant}\quad\Longrightarrow\quad\begin{cases}\ker T=V,&T=0,\\\ker T=0,\ \operatorname{im}T=W,&T\ne0.\end{cases} $$

The reusable statement

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad T:V\to W,\qquad T\rho_V(g)=\rho_W(g)T,\\[5pt] \mathsf{C}\;&:\quad V,W\ \text{irreducible}\Longrightarrow T=0\ \text{or}\ T\ \text{is an isomorphism}. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] V,W\ \text{irreducible}\Longrightarrow T=0\ \text{or}\ T\ \text{is an isomorphism} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 19 October 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.