A representation turns group elements into linear transformations. This allows group structure to be studied with eigenspaces, invariant subspaces, traces, and decompositions.
Linear representations
Definition. A representation of a group \(G\) over a field \(F\) is a homomorphism \[ \rho:G\to\operatorname{GL}(V), \] where \(V\) is an \(F\)-vector space. The dimension of the representation is \(\dim_F V\).
Example. The sign representation of \(S_n\) sends a permutation to multiplication by \(1\) or \(-1\). The permutation representation of \(S_n\) on \(F^n\) permutes the standard basis vectors.
Invariant subspaces and irreducibility
Definition. A subspace \(W\subset V\) is \(G\)-invariant if \(\rho(g)W\subset W\) for all \(g\in G\). A nonzero representation is irreducible if its only invariant subspaces are \(0\) and \(V\).
Example. The permutation representation of \(S_n\) on \(F^n\) contains the invariant line spanned by \((1,\ldots,1)\). The quotient by this line, or the complementary hyperplane \(\sum x_i=0\) when allowed, gives the standard representation.
Subrepresentations and quotient representations
Proposition. If \(W\subset V\) is invariant, then \(W\) is a representation of \(G\), and \(V/W\) becomes a quotient representation by \(g\cdot(v+W)=gv+W\).
Proof. Invariance ensures that the action restricts to \(W\) and that changing \(v\) by an element of \(W\) changes \(gv\) by an element of \(W\). The group action axioms descend from those on \(V\).
Maschke theorem
Theorem. If \(G\) is finite and \(\operatorname{char}F\nmid |G|\), then every finite-dimensional representation of \(G\) over \(F\) is completely reducible: it is a direct sum of irreducible representations.
Proof. Given an invariant subspace \(W\subset V\), choose any projection \(P:V\to W\). Average it over the group: \[ P_G=\frac{1}{|G|}\sum_{g\in G}\rho(g)P\rho(g)^{-1}. \] Then \(P_G\) is \(G\)-equivariant and projects onto \(W\), so \(\ker P_G\) is an invariant complement.
Regular representation
Definition. The regular representation of a finite group \(G\) is the representation on the vector space \(F[G]\) with basis \(\{e_g:g\in G\}\), where \(h\cdot e_g=e_{hg}\).
Remark. Over a suitable field, the regular representation contains every irreducible representation, with multiplicity equal to its dimension.
Representation as module over a group algebra
Proposition. Representations of \(G\) over \(F\) are the same as left modules over the group algebra \(F[G]\).
Maschke decomposition
Maschke's theorem. If \(G\) is finite and \(\operatorname{char}F\nmid|G|\), every finite-dimensional \(F\)-representation of \(G\) is a direct sum of irreducible representations.
Proof. Let \(W\subset V\) be invariant and choose any projection \(P:V\to W\). Average it: \[ \overline P=\frac1{|G|}\sum_{g\in G}\rho(g)P\rho(g)^{-1}. \] Then \(\overline P\) is \(G\)-equivariant and still restricts to the identity on \(W\). Hence \[ V=W\oplus\ker\overline P \] as representations. Induct on dimension.
The characteristic assumption is sharp. For the cyclic group \(C_p\) over \(\mathbb F_p\), the regular representation contains a nontrivial invariant line without an invariant complement; averaging is impossible because \(|G|=0\) in the field.
The regular representation
Theorem. Over \(\mathbb C\), the regular representation decomposes as \[ \mathbb C[G]\cong\bigoplus_{\pi\in\widehat G} (\dim\pi)\,\pi. \] Consequently \[ \sum_{\pi\in\widehat G}(\dim\pi)^2=|G|. \]
Proof. The character of the regular representation is \(|G|\) at the identity and zero elsewhere. Its inner product with an irreducible character \(\chi_\pi\) is \[ \frac1{|G|}\sum_g\chi_{\mathrm{reg}}(g) \overline{\chi_\pi(g)}=\chi_\pi(e)=\dim\pi. \] Taking dimensions yields the final identity.