What are the irreducible representations of finite abelian groups over \(\mathbb{C}\)? Prove that every irreducible representation is 1-dimensional, and describe the dual group \(\widehat{G} \cong G\).
Solution
Let \((\rho,V)\) be an irreducible complex representation of a finite abelian group \(G\). For every \(g\in G\), the operator \(\rho(g)\) commutes with every \(\rho(h)\). By Schur’s lemma, each \(\rho(g)\) is scalar. Hence every one-dimensional subspace of \(V\) is \(G\)-stable, so irreducibility forces \(\dim_{\mathbb C}V=1\).
Thus the irreducible representations are exactly the characters \[ \widehat G=\operatorname{Hom}(G,\mathbb C^\times), \] with pointwise multiplication. Because \(G\) is finite, all character values are roots of unity.
For \(C_n=\langle g\rangle\), a character is determined by the choice of \(\chi(g)\) among the \(n\)th roots of unity, so \(\widehat{C_n}\cong C_n\). If \[ G\cong C_{n_1}\times\cdots\times C_{n_r}, \] then \[ \widehat G\cong \widehat{C_{n_1}}\times\cdots\times\widehat{C_{n_r}}\cong G. \] This isomorphism with \(G\) is noncanonical; the evaluation map \(G\to\widehat{\widehat G}\) is canonical.