Representations of Finite Groups (over $\mathbb{C}$)¶

Job: complete reducibility, characters, and orthogonality — the character table as a complete invariant of the semisimple algebra $\mathbb{C}[G]$.

Core¶

Definitions¶

A representation of a finite group $G$ on a finite-dimensional $\mathbb{C}$-vector space $V$ is a homomorphism $\rho:G\to\operatorname{GL}(V)$, equivalently a left $\mathbb{C}[G]$-module structure on $V$. Degree $=\dim V$. Morphism of reps: linear $T:V\to W$ with $T\rho(g)=\sigma(g)T$ (intertwiner).

Notion	Meaning
Irreducible (simple)	no invariant subspace $0\subsetneq W\subsetneq V$
Completely reducible	$V\cong\bigoplus V_i$ irreducible
Character $\chi_V$	$\chi_V(g)=\operatorname{tr}\rho(g)$; class function
Regular representation	$\mathbb{C}[G]$ acting on itself by left mult.

Isomorphic reps $\Leftrightarrow$ same character (over $\mathbb{C}$, finite $G$): $\chi$ determines the multiset of irreducible constituents.

Maschke and complete reducibility¶

Theorem (Maschke). Let $G$ be finite, $k$ a field with $\operatorname{char} k\nmid |G|$ (e.g. $k=\mathbb{C}$). Then every short exact sequence of $k[G]$-modules splits: every representation is completely reducible.

Proof sketch. Average any projection onto a subrep $W$ over $G$: $P\mapsto \frac{1}{|G|}\sum_g \rho(g)P\rho(g)^{-1}$ yields a $G$-equivariant projection.

Corollary. $\mathbb{C}[G]$ is semisimple Artinian. By Artin–Wedderburn: $$ \mathbb{C}[G]\cong\prod_{i=1}^r M_{n_i}(\mathbb{C}),\qquad n_i=\dim V_i, $$ one matrix factor per irrep $V_i$ up to iso. Number of irreps $r=$ number of conjugacy classes.

Dimension formula: $$ \sum_i (\dim V_i)^2 = |G|. $$ Regular representation: $\chi_{\mathrm{reg}}(1)=|G|$, $\chi_{\mathrm{reg}}(g)=0$ for $g\neq 1$; contains each irrep $V_i$ with multiplicity $\dim V_i$.

Characters and orthogonality¶

$\chi_V$ is a class function: $\chi(hgh^{-1})=\chi(g)$. Space of class functions has dim $=$ # conjugacy classes, orthonormal basis the irreducible characters.

Inner product: $$ \langle\chi,\psi\rangle = \frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}. $$ (Over $\mathbb{C}$, $\chi(g^{-1})=\overline{\chi(g)}$ for finite-order unitary images; equivalently $\chi(g^{-1})$ without conj. if careful.)

Theorem	Statement
Schur orthogonality (rows)	$\langle\chi_i,\chi_j\rangle=\delta_{ij}$ for irreps
Column orthogonality	$\sum_i\chi_i(g)\overline{\chi_i(h)}=
Multiplicity	$V\cong\bigoplus_i V_i^{\oplus m_i}$ with $m_i=\langle\chi_V,\chi_i\rangle$
Irreducibility test	$\chi$ irreducible $\Leftrightarrow\langle\chi,\chi\rangle=1$

Schur’s lemma. $\operatorname{Hom}_G(V,W)=0$ if $V\not\cong W$ irreps; $\operatorname{End}_G(V)=\mathbb{C}\cdot\mathrm{id}$ if $V$ irrep (alg. closed field).

Constructions¶

Operation	Character
$V\oplus W$	$\chi_V+\chi_W$
$V\otimes W$	$\chi_V\chi_W$ (pointwise)
$V^*$	$\chi(g^{-1})$ ( $=\overline{\chi(g)}$ for finite $G/\mathbb{C}$)
$\operatorname{Hom}(V,W)\cong V^*\otimes W$	product as above
$\operatorname{Ind}_H^G W$	formula $\chi_{\mathrm{Ind}}(g)=\sum_{t\in T,\,t^{-1}gt\in H}\chi_W(t^{-1}gt)$
$\operatorname{Res}_H^G V$	restriction of $\chi$

Frobenius reciprocity: $\langle\operatorname{Ind}_H^G\psi,\chi\rangle_G=\langle\psi,\operatorname{Res}_H^G\chi\rangle_H$.

Character table¶

Rows: irreps; columns: conjugacy classes; entries $\chi_i(g_j)$. Unitary (up to scaling by class sizes) by orthogonality. Determines: kernel of $\rho$ (where $\chi(g)=\chi(1)$), whether $V$ is real/complex/quaternionic (Frobenius–Schur indicator $\frac{1}{|G|}\sum\chi(g^2)$), abelianization ($1$-dimensional reps $\leftrightarrow G^{\mathrm{ab}}$).

Abelian $G$: all irreps $1$-dimensional; $\widehat{G}\cong G$ non-canonically for finite abelian; $\mathbb{C}[G]\cong\prod_{|G|}\mathbb{C}$.

See it¶

$S_3$ and $S_4$. Character table of $S_3$: trivial, sign, standard $2$-dim; classes $\mathrm{id}$, transpositions, $3$-cycles. For $S_4$: five classes; irreps include standard $3$-dim on $\sum x_i=0$ and its tensor with sign. Computing $\langle\chi_{\mathrm{std}}^{\otimes 2},\chi_i\rangle$ decomposes tensors — template for all finite reflection groups and McKay correspondence (finite subgroups of $\operatorname{SU}(2)$).

Burnside $p^a q^b$ theorem (character-theoretic proof). Nontrivial irrep degree divisible by prime + integrality of character values $\Rightarrow$ non-simple groups of order $p^aq^b$. Modern finite group theory still uses character estimates (e.g. simple group classification ingredients).

Check¶

Prove that if $\chi$ is irreducible of degree $n$ and $z\in Z(G)$, then $\rho(z)=\omega(z)I$ with $\omega(z)^n=\chi(z)$ a root of unity.

Schur: $\rho(z)$ intertwines $\rho$ with itself, hence scalar $\omega(z)I$. Then $\chi(z)=n\omega(z)$; $\rho(z)^{|G|}=I\Rightarrow\omega(z)$ root of unity. Also $\omega:Z(G)\to\mathbb{C}^\times$ is a linear character.

Give a counterexample to Maschke when $\operatorname{char} k\mid|G|$.

$G=\mathbb{Z}/p$, $k=\mathbb{F}_p$, $\rho(1)=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ on $k^2$: unique invariant line, nonsplit extension of trivial by trivial. Or: augmentation submodule of $k[G]$ is nilpotent ideal, so $k[G]$ not semisimple.

Pitfalls¶

Characters are not homomorphisms unless degree $1$; $\chi(gh)\neq\chi(g)\chi(h)$ in general.
Same character values on generators do not define a character without extending by conjugacy and relations.
Modular representation theory ($\operatorname{char} k\mid|G|$): Brauer characters, decomposition matrices — ordinary character table does not classify indecomposables.
Infinite discrete groups: complete reducibility fails; unitary reps on Hilbert space are a different subject.

Refs¶

Serre, Linear Representations of Finite Groups
Fulton–Harris, Representation Theory (Part I)
Isaacs, Character Theory of Finite Groups
James–Liebeck, Representations and Characters of Groups

Next¶

Lie algebras · Lie groups · RT index · Group theory · Modules · Homological algebra

Operation	Character
\(V\oplus W\)	\(\chi_V+\chi_W\)
\(V\otimes W\)	\(\chi_V\chi_W\) (pointwise)
\(V^*\)	\(\chi(g^{-1})\) ( \(=\overline{\chi(g)}\) for finite \(G/\mathbb{C}\))
\(\operatorname{Hom}(V,W)\cong V^*\otimes W\)	product as above
\(\operatorname{Ind}_H^G W\)	formula \(\chi_{\mathrm{Ind}}(g)=\sum_{t\in T,\,t^{-1}gt\in H}\chi_W(t^{-1}gt)\)
\(\operatorname{Res}_H^G V\)	restriction of \(\chi\)

Representations of Finite Groups (over \(\mathbb{C}\))¶