# Definition The regular representation is a [[Irreducible Representation|reducible matrix representation]] obtained by rearranging the [[Group Multiplication Table|multiplication table]] so that the identity element is always on the diagonal. When this is done, the [[Group Element|group elements]] label the columns and their inverses label the rows. Then, we take the table and turn it into a matrix. The matrix for the element $g$ has zeros everywhere except where the element $g$ occurs where it has a $1$. It can be proved that this construction does indeed form a [[group representation]]. The matrices of the regular representation are denoted by $D^{\text{reg}}$. Note that, by construction of the regular representation, the [[Character of a Representation|character]] of the representations of all [[group element|group elements]] is zero except, i.e. $\chi^{\text{reg}}(g) = 0 \forall g \in G$ except the identity element which has $\chi^{\text{reg}}(e) = h$ where $h$ is the order of the [[group]]. # Theorem The regular representation contains each [[Irreducible Representation]] a number of times equal to the [[Dimension|dimensionality]] of the representation. ### Proof We know from [[The Decomposition Theorem for Reducible Representations]] that we can write $\chi^{\text{reg}}(\mathcal{C}_k)$ as: $ \chi^{\text{reg}}(\mathcal{C}_k) = \sum_{\Gamma_i} \alpha_i \chi^{(\Gamma_i)}(\mathcal{C}_k) $ where the sum is over the [[Irreducible Representation|irreducible representations]] and the $\alpha_i$ is the unique coefficient (as previously shown), and is given by $ \alpha_i = \frac{1}{h} \sum_k N_k [\chi^{(\Gamma_i)}(\mathcal{C}_k)]^* \chi^{\text{reg}}(\mathcal{C}_k). $ Recall that $\alpha_i$ is the number of times the reducible representation contains the [[Irreducible Representation]]. Observe that $N_e =1$, i.e., the identity element forms a [[Conjugacy Class|class]] by itself and thus its class has only 1 element (as we already know). Moreover, note that the character of the class of the identity in any [[Irreducible Representation]] is given by the [[trace]] of the identity matrix, i.e., $\chi^{(\Gamma_i)}(e) = \ell_i$. Using the information given in the [[#Definition|definition]] about the [[Character of a Representation|character]] of the different elements, we can compute $\alpha_i$ as: $ \alpha_i = \frac{1}{h}(N_e \cdot \chi^{\text{reg}}(\mathcal{C}_e)\cdot \chi^{(\Gamma_i)}(\mathcal{C}_e)) + 0 = \frac{1}{h} (1 \cdot h \cdot \ell_i) = \ell_i $ So we finally end up with: $ \alpha_i = \ell_i. \quad \square $ Note that, since $\ell_i > 0$, then the regular representation contains each [[Irreducible Representation]] at least once! On the surface, this might seem to be helpful. However, it is very difficult and tedious in practice to extract the irreducible representations from the regular representation.