Square sum of irrep dimensions

As an immediate consequence of orthogonality of irreps, it follows that the square sum of the dimensions of all (non-equivalent) irreps of a finite group $𝐺$ equals the order $| 𝐺 |$ of the group. #m/thm/rep

\sum 𝛾 \in ˆ 𝐺 (𝑑 𝛾) 2 = | 𝐺 |

Corollary

Since Irreps of abelian groups are 1-dimensional, it follows that the number of irreps $∣ ˆ 𝐺 ∣$ of an abelian group $𝐺$ equals the order of the group $| 𝐺 |$ . #m/thm/rep

𝑍 (𝐺) = 𝐺 ⟹ ∣ ˆ 𝐺 ∣ = | 𝐺 |

#state/tidy | #lang/en | #SemBr