A representation of a finite group \(G\) over a field \(k\) is a finite-dimensional left \(k[G]\)-module. A module is semisimple if it is a direct sum of simple modules. Maschke’s theorem: if the characteristic of \(k\) does not divide \(\abs{G}\), every \(k[G]\)-module is semisimple. By the Artin–Wedderburn theorem, a semisimple ring is a finite product of matrix rings \(M_{n_i}(D_i)\) over division rings \(D_i\). Schur’s lemma: over an algebraically closed field, the \(G\)-equivariant endomorphisms of an irreducible representation are the scalars. Complex characters of \(G\) are class functions, and the number of irreducible complex characters of \(G\) equals the number of conjugacy classes.
Artin--Wedderburn theorem (Tag T-ZOTWN)
Let \(R\) be a nonzero unital ring that is semisimple as a left module over itself. Then there are integers \(m\geq 1\) and \(n_1,\ldots,n_m\geq 1\) and division rings \(D_1,\ldots,D_m\) such that \[ R \cong \bigoplus_{i=1}^m \Mat_{n_i}(D_i), \] a finite direct sum of the rings of \(n_i\times n_i\) matrices over \(D_i\).
Semisimple rings are finite direct sums of matrix rings over division rings.
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.
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.
Character of a representation (Tag D-CTAKB)
Let \(G\) be a group, let \(k\) be a field, and let \(M\) be a finite-dimensional representation of \(G\) over \(k\), that is, a finite-dimensional \(k\)-vector space with a linear action \(G\actson M\). For \(g\in G\), let \[ T_g\colon M \to M, \qquad m \mapsto g\actson m. \] The character of \(M\) is the function \(\chi_M\colon G\to k\), \(\chi_M(g) \coloneqq \tr T_g\).