Lagmental Vicfred

Character Inner Products Count Multiplicities by Vicfred

tags: mathematics, representation theory, characters, group algebras

Over the complex numbers, the inner product of characters extracts the multiplicity of an irreducible constituent. I will separate the object being defined from the consequence being claimed.

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.

$$ \langle\chi,\psi\rangle_G=\frac1{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ V\cong\bigoplus_i m_iV_i\Longrightarrow m_i=\langle\chi_V,\chi_i\rangle_G $$

Derivation

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

$$ \begin{array}{c|ccc}&e&(123)&(12)\\\hline\chi_{\mathrm{triv}}&1&1&1\\\chi_{\mathrm{sgn}}&1&1&-1\\\chi_{\mathrm{std}}&2&-1&0\end{array} $$

Invariant content

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \langle\chi,\psi\rangle_G=\frac1{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)},\\[5pt] \mathsf{C}\;&:\quad V\cong\bigoplus_i m_iV_i\Longrightarrow m_i=\langle\chi_V,\chi_i\rangle_G. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] V\cong\bigoplus_i m_iV_i\Longrightarrow m_i=\langle\chi_V,\chi_i\rangle_G \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 10 April 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.