Regular representation

Let $G$ be a finite group and $k$ a field.

Definition (left regular representation)

The left regular representation of $G$ over $k$ is the representation

on the group algebra $k[G]$, where $k[G]$ is the $k$-vector space with basis $\{e_g : g\in G\}$ and the action is

Equivalently, $G$ acts by left multiplication on $k[G]$.

(There is also a right regular representation $e_g\mapsto e_{gh^{-1}}$, which is generally different but closely related.)

Character of the regular representation (over $\mathbb C$)

Over $\mathbb C$, the character $\chi_{\mathrm{reg}}$ of the regular representation satisfies

Reason: $\lambda(g)$ permutes the basis $\{e_h\}$; it has fixed points iff $g=1$, and the trace of a permutation matrix equals the number of fixed basis vectors.

Decomposition over $\mathbb C$

If $\{V_i\}$ runs over the irreducible representations of $G$ over $\mathbb C$, with $\dim V_i = d_i$, then the regular representation decomposes as

In particular,

which is the content of the sum of squares formula . The index $i$ ranges over exactly as many irreducibles as there are conjugacy classes (see number of irreducibles equals number of conjugacy classes ).

Examples

1. Cyclic group $C_n$ over $\mathbb C$.
   All irreducibles of $C_n$ are 1-dimensional characters $\chi_j(g)=\zeta_n^{j}$ (for $j=0,\dots,n-1$). Hence

   $$\mathbb C[C_n]\;\cong\;\bigoplus_{j=0}^{n-1}\chi_j,$$

   i.e. the regular representation splits into $n$ distinct 1-dimensional representations.

2. $S_3$ over $\mathbb C$.
   The irreducible degrees are $1,1,2$ (trivial, sign, standard). Therefore

   $$\mathbb C[S_3] \;\cong\; \mathbf{1}\;\oplus\;\mathrm{sgn}\;\oplus\;2\cdot V_{\mathrm{std}}.$$

   The multiplicity of each irreducible equals its dimension.

3. Dihedral group $D_4$ of order $8$ over $\mathbb C$.
   $D_4$ has four 1-dimensional irreducibles and one 2-dimensional irreducible $V$. Thus

   $$\mathbb C[D_4] \;\cong\; \chi_1\oplus\chi_2\oplus\chi_3\oplus\chi_4 \;\oplus\; 2\cdot V,$$

   and $8 = 1^2+1^2+1^2+1^2+2^2$.