The Left Regular Representation

# The Left Regular Representation

 Definition: Let $G$ be a finite group. Let $\mathbb{C}[G]$ be the space of all linear combinations $\sum_{g \in G} c_g g$ where $c_g \in \mathbb{C}$ for each $g \in G$.
 Definition: Let $G$ be a finite group. The Left Regular Representation of $G$ on $\mathbb{C}[g]$ is defined to be the group action defined for all $g \in G$ and all $\displaystyle{\sum_{h \in G} c_h h \in \mathbb{C}[G]}$ by $\displaystyle{g \cdot \sum_{h \in G} c_h h := \sum_{h \in G} c_h (gh) = \sum_{x \in G} c_{g^{-1}x} x}$. The Character of the Left Regular Representation is denoted by $\chi_{\mathrm{reg}}$.

The character of the left regular representation of $G$ is given by:

(1)
\begin{align} \chi_{\mathrm{reg}}(g) = \left\{\begin{matrix} |G| & \mathrm{if} \: g = e \\ 0 & \mathrm{if} \: g \neq e \end{matrix}\right. \end{align}
 Proposition 1: Let $G$ be a (nontrivial) finite group. Then the left regular representation of $G$ is reducible.
• Proof: We have that:
(2)
\begin{align} \quad \langle \chi_{\mathrm{reg}}, \chi_{\mathrm{reg}} \rangle = \frac{|G|^2}{|G|} = |G| \neq 1 \end{align}
 Corollary 2: Let $G$ be a finite group. Then every irreducible representation of $G$ is a subrepresentation of the left regular representation.
 Proposition 3: Let $G$ be a finite group. If $V$ is an irreducible representation of $G$ then the multiplicity of $V$ in the left regular representation of $G$ is $\mathrm{dim}(V)$.
• Proof: By one of the corollaries on the page mentioned above, we know that this multiplicity is given by:
(3)
\begin{align} \quad \langle \chi_V, \chi_{\mathrm{reg}} \rangle = \frac{1}{|G|} \chi_V(e) \chi_{\mathrm{reg}}(e) = \frac{1}{|G|} \mathrm{dim}(V) |G| = \mathrm{dim}(V) \quad \blacksquare \end{align}