Representations: Maschke and Schur

A representation makes a group concrete by having it act linearly on a vector space — every group element becomes a matrix, and composition becomes matrix multiplication. Representation theory is where group theory meets linear algebra, and it is the language in which physical symmetries act on state spaces. This page covers the finite-dimensional basics; character-theory.md adds the computational engine, and reps-lie.md carries the story to Lie algebras. It specialises the general group actions to linear ones.

Linear representations

A representation of $G$ on a vector space $V$ over a field $\mathbb{F}$ (here $\mathbb{C}$ unless noted) is a homomorphism $$\rho :G\to GL(V),$$ the group of invertible linear maps on $V$. Equivalently $G$ acts on $V$ with each $g$ acting linearly: $$\rho (g_1{g}_2)=\rho (g_1)\rho (g_2),\rho (e)=1_V.$$ The dimension (or degree) of the representation is $\dim V$. Choosing a basis turns $\rho$ into a homomorphism $G\to GL(n,\mathbb{F})$ — a matrix representation.

• $\rho$ is faithful if injective (the group acts without collapse).
• $\rho$ is unitary if $V$ carries an inner product preserved by every $\rho (g)$, i.e. $\rho (g{)}^{†}\rho (g)=1$. Physical Hilbert-space symmetries must be unitary (probabilities are conserved).
• The trivial representation sends every $g\mapsto 1$ on $V=\mathbb{F}$.
• The regular representation is the action of $G$ on the vector space $\mathbb{F}[G]$ with basis $\{e_g{\}}_{g\in G}$ by left multiplication; it contains every irreducible (see below and character-theory.md).

Two representations $\rho ,{\rho }^{\prime }$ on $V,V^{\prime }$ are equivalent (isomorphic) if there is an invertible intertwiner $T:V\to V^{\prime }$ with $T\rho (g)={\rho }^{\prime }(g)T$ for all $g$ — a change of basis conjugating one into the other.

Reducibility

A subspace $W\subseteq V$ is invariant (a subrepresentation) if $\rho (g)W\subseteq W$ for all $g$. The representation is:

• irreducible (an irrep) if its only invariant subspaces are $\{0\}$ and $V$;
• reducible otherwise;
• completely reducible (semisimple) if $V$ decomposes as a direct sum of irreducible subrepresentations.

Irreps are the atoms; the central problem is to classify them and decompose arbitrary representations into them.

Maschke's theorem

Maschke's theorem. If $G$ is finite and $\mathrm{char}\mathbb{F}\nmid |G|$ (in particular over $\mathbb{C}$), then every finite-dimensional representation of $G$ is completely reducible.

Idea. Given an invariant subspace $W$, pick any complement and average its projection over the group, $$P=\frac{1}{|G|}\sum _{g\in G}\rho (g)P_0\rho (g{)}^{-1},$$ producing a $G$-invariant projection whose kernel is an invariant complement. The same averaging trick makes any representation of a finite group unitary for a suitable inner product. (Averaging over a finite group is replaced by integrating against Haar measure for compact groups — see compact-groups.md — which is why the compact-group representation theory looks identical.)

The hypothesis matters: in modular representation theory ($\mathrm{char}\mathbb{F}\mid |G|$) complete reducibility fails, and the subject becomes genuinely harder.

Schur's lemma

Schur's lemma. Let $\rho ,{\rho }^{\prime }$ be irreducible and $T$ an intertwiner. (i) $T$ is either $0$ or an isomorphism. (ii) Over an algebraically closed field (e.g. $\mathbb{C}$), an intertwiner from an irrep to itself is a scalar: $T=\lambda 1$.

Consequences.

• The only operators commuting with an irreducible action are scalars. Hence any operator that commutes with all $\rho (g)$ — a Casimir, say — acts as a constant on each irrep, providing an irrep label. This is exactly how Casimir eigenvalues label irreps of Lie algebras in reps-lie.md.
• Irreps of an abelian group are all $1$-dimensional (every $\rho (g)$ commutes with the whole image, so is scalar). Their characters are the group's Pontryagin dual, central to cyclic-abelian.md.
• Schur's lemma underlies the orthogonality relations that make character theory computable.

Operations on representations

New representations are built from old:

• Direct sum $\rho \oplus {\rho }^{\prime }$ on $V\oplus V^{\prime }$ (block-diagonal matrices).
• Tensor product $\rho \otimes {\rho }^{\prime }$ on $V\otimes V^{\prime }$, with $(\rho \otimes {\rho }^{\prime })(g)=\rho (g)\otimes {\rho }^{\prime }(g)$. Decomposing a tensor product into irreps is the Clebsch–Gordan problem — for $SU(2)$ this is the addition of angular momenta.
• Dual (contragredient) ${\rho }^{\ast }$ on $V^{\ast }$ by ${\rho }^{\ast }(g)=\rho (g^{-1}{)}^T$.
• Hom / adjoint representations on $\mathrm{Hom}(V,V^{\prime })$.

Projective representations

Quantum mechanics needs a weaker notion: physical states are rays, so a symmetry need only be represented up to phase. A projective representation satisfies $$\rho (g_1)\rho (g_2)=\omega (g_1,g_2)\rho (g_1{g}_2)$$ for phases $\omega \in U(1)$. Projective representations of $G$ correspond to ordinary representations of a central extension of $G$ — for Lie groups, of the universal cover $\tilde{G}$. This is why $SU(2)$ (not $SO(3)$) and $SL(2,\mathbb{C})$ (not the Lorentz group) are the groups that act in quantum theory; the mechanism is spelled out in topology-covers.md.

References

• Fulton & Harris, Representation Theory: A First Course, Part I.
• Serre, Linear Representations of Finite Groups — the concise classic.
• Dummit & Foote, Abstract Algebra, Ch. 18.