By Maschke’s theorem, every finite-dimensional complex representation of a finite group is a direct sum of irreducible representations; by Schur’s lemma, a homomorphism of representations between irreducible representations is zero or an isomorphism, and over \(\CC\) every endomorphism of an irreducible representation is a scalar.
Group ring (Tag D-FHUV5)
Let \(R\) be a commutative ring and \(G = \ts{g_1, \ldots, g_n}\) a finite group. The group ring \(RG\) is the free \(R\)-module with basis the elements of \(G\), \[ RG\coloneqq\ts{ \sum_{i=1}^n a_i g_i \st a_i\in R }, \] with the multiplication that extends the group law \(R\)-bilinearly: \((ag)(bh)\coloneqq(ab)(gh)\) for \(a, b\in R\) and \(g, h\in G\).
The group ring \(RG\) is a ring with identity \(1_R 1_G\). If \(R\neq 0\), then \(RG\) is commutative if and only if \(G\) is abelian.
See [@DF04].
Maschke’s theorem
Maschke's theorem (Tag T-PIO2B)
Let \(G\) be a finite group and \(k\) a field with \(\ch(k)\nmid \size G\). Then every finite-dimensional representation of \(G\) over \(k\) is a direct sum of irreducible representations.
If \(\ch(k)\nmid \abs{G}\), finite-dimensional \(k[G]\)-modules are direct sums of simple \(k[G]\)-modules.
(Hypotheses and proof)
Let \(G\) be a finite group and \(k\) a field whose characteristic does not divide \(\size G\). Every finite-dimensional representation of \(G\) over \(k\) is a direct sum of irreducible representations, so \(k[G]\) is semisimple, and a finite-dimensional complex representation is determined up to isomorphism by the multiplicities of its irreducible summands.
If \(\operatorname{char}k = p\) divides \(\size G\), the conclusion can fail: the regular representation of \(\ZZ/p\) over \(\FF_p\) is indecomposable and not irreducible. If \(G\) is infinite, the conclusion can fail: \(\ZZ\) acting on \(\CC^2\) by \(n\mapsto\matt{1}{n}{0}{1}\) has the invariant line \(\CC e_1\) and no invariant complement.
For a subrepresentation \(W\subseteq V\), choose any linear projection \(\pi\colon V\to W\) and set \(\pi_G \da {1\over\size G}\sum_{g\in G} g\pi \inverseof{g}\). Then \(\pi_G\) is a \(G\)-equivariant projection onto \(W\), and \(\ker\pi_G\) is a \(G\)-invariant complement of \(W\); the division by \(\size G\) uses the hypothesis on the characteristic.
Schur’s lemma
Schur's lemma (Tag T-YHH3M)
Let \(G\) be a group, \(k\) an algebraically closed field, and \(M\) an irreducible representation of \(G\) over \(k\) with \(\dim_k M < \infty\). Then every \(G\)-equivariant endomorphism of \(M\) is multiplication by a scalar, so the map \[ k \mapsvia{\sim} \Endo_G(M),\qquad \lambda\mapsto \lambda\cdot\id_M, \] is an isomorphism.
For an irreducible representation \(M\) of \(G\) over an arbitrary field \(k\), every nonzero \(G\)-equivariant endomorphism of \(M\) has \(G\)-stable kernel \(0\) and image \(M\), so \(\Endo_G(M)\) is a division algebra over \(k\).
Over an algebraically closed field, an irreducible finite-dimensional representation has only scalar equivariant endomorphisms.
(The two statements)
For irreducible representations \(V\) and \(W\) of \(G\):
every \(G\)-equivariant linear map \(V\to W\) is zero or an isomorphism;
if \(V\) is finite-dimensional over an algebraically closed field such as \(\CC\), every \(G\)-equivariant linear map \(f\colon V\to V\) is a scalar multiple of the identity.
For the second statement, \(f\) has an eigenvalue \(\lambda\) because the field is algebraically closed, and \(f - \lambda I\) is a \(G\)-equivariant map with nonzero kernel, hence zero by the first statement.
For irreducible complex representations, \(\dim \Hom_G(V,W)=0\) if \(V\not\cong W\) and \(\dim \operatorname{End}_G(V)=1\); together with Maschke’s theorem, these give the character orthogonality relations and the multiplicity formula.
Consequences
Let \(V_1,\ldots,V_r\) be the irreducible complex representations of \(G\) up to isomorphism, with \(d_i = \dim V_i\). Then \(\CC[G] \cong \bigoplus_i \operatorname{End}(V_i) \cong \bigoplus_i \Mat_{d_i}(\CC)\), so \[ \size G = \sum_i d_i^2, \] and \(r\) equals the number of conjugacy classes of \(G\).
\(S_3\) has three conjugacy classes and order \(6\); its two degree-one representations are the trivial and sign representations, so \(6 = 1 + 1 + d_3^2\) gives \(d_3=2\).