Sum of squares of degrees - Imported Knowlpedia Content

Let $G$ be a finite group and let $k$ be an algebraically closed field with $\operatorname{char}(k)\nmid |G|$ (in particular, $k=\mathbb C$ ). Let

\{V_1,\dots,V_r\}

be a complete set of pairwise non-isomorphic finite-dimensional irreducible representations of $G$ over $k$ , and write $d_i=\dim_k(V_i)$ .

Theorem (sum of squares of degrees)

\sum_{i=1}^r d_i^2 \;=\; |G|.

Equivalently, the underlying $kG$ -module of the regular representation decomposes as

k[G]\;\cong\;\bigoplus_{i=1}^r d_i\,V_i,

and taking dimensions gives $\dim_k(k[G])=|G|=\sum_i d_i\dim(V_i)=\sum_i d_i^2$ .

This decomposition is a standard consequence of Maschke-type semisimplicity statements (so $k[G]$ is semisimple) together with multiplicity computations using Schur's lemma.

Examples

Cyclic group $C_n$ . Over $\mathbb C$ , every irreducible representation of $C_n$ is $1$ -dimensional (a character). There are $n$ such characters, so $\sum_i d_i^2 \;=\; \underbrace{1^2+\cdots+1^2}_{n\text{ times}} \;=\; n \;=\; |C_n|.$

Symmetric group $S_3$ (order $6$ ). $S_3$ has irreducible degrees $1,1,2$ (trivial, sign, and the $2$ -dimensional standard representation), hence $1^2+1^2+2^2 \;=\; 6 \;=\; |S_3|.$

Dihedral group $D_8$ (symmetries of a square, order $8$ ). $D_8$ has four $1$ -dimensional irreducibles and one $2$ -dimensional irreducible, so $4\cdot 1^2 + 1\cdot 2^2 \;=\; 8 \;=\; |D_8|.$