Regular representation

The regular representation of a group is a reducible representation that is generated from a rearranged multiplication table of the group. It is used to derive an important property (see eq40 below) for constructing character tables.

Consider the re-arranged multiplication table for the $C_{3v}$ point group such that all the identity elements are along the diagonal:

An element of the regular representation of the group $G$ is derived from table II in the form of a $\vert G\vert\times\vert G\vert$ matrix, whose entries are 1 when the element of $G$ occurs in table II and zero otherwise. For example,

In other words, we have

$\Gamma_R(a_i)_{jk}=\begin{cases}1&if\;a_j^{-1}a_k=a_i\\0&otherwise\end{cases}\;\;\;\;\;\;\;\;31$

where $\Gamma_R(a_i)_{jk}$ is the $j$ -th row and $k$ -th column matrix entry of the $\alpha_i$ -th element of the $R$ representation of $G$ .

Question

Show that each matrix of the regular representation of the $C_{3v}$ point group has an inverse.

Answer

The entry 1 appears only once in every row (or column) of a matrix, e.g. $\Gamma_R(C_3)$ , of the regular representation. We can therefore swap the rows (or columns) of $\Gamma_R(C_3)$ to form $\Gamma_R(E)$ . According to determinant property 1, $\vert\Gamma_R(E)\vert=1\neq 0$ and consequently, according to determinant property 8, $\vert\Gamma_R(C_3)\vert\neq 0$ . Therefore, $\Gamma_R(C_3)$ is non-singular, according to determinant property 11. The same logic applies to the rest of the matrices.

If these derived matrices are truly elements of a representation of $G$ , then they must satisfy the closure property of $G$ , i.e.

$\sum_k\Gamma_R(a_i)_{jk}\Gamma_R(a_m)_{kl}=\Gamma_R(a_p)_{jl}\;\;\;only\;if\;a_p=a_ia_m\;\;\;\;\;\;\;\;32$

where

$\Gamma_R(a_m)_{kl}=\begin{cases}1&if\;a_k^{-1}a_l=a_m\\0&otherwise\end{cases}\;\;\;\;\;\;\;\;34$

$\Gamma_R(a_p)_{jl}=\begin{cases}1&if\;a_j^{-1}a_l=a_p\\0&otherwise\end{cases}\;\;\;\;\;\;\;\;35$

Therefore, $a_j^{-1}a_l$ in eq35 needs to be equal to $a_ia_m$ to satisfy eq32. To prove this, we multiply the condition $a_j^{-1}a_k=a_i$ in eq31 on the left by $a_j$ to give $a_k=a_ja_i$ . Similarly, we multiply the condition $a_k^{-1}a_l=a_m$ in eq34 on the left by $a_k$ and then on the right by $a_m^{-1}$ to give $a_la_m^{-1}=a_k$ . Combining the two results, we have $a_ja_i=a_la_m^{-1}$ , which when multiplied on the right by $a_m$ and then on the left by $a_j^{-1}$ , gives $a_ia_m=a_j^{-1}a_l$ , which completes the proof.

By inspecting table II, the character of the regular representation is

$\chi_R(a_i)=\sum_{k=1}^{\vert G\vert}\Gamma_R(a_i)_{kk}=\begin{cases}\vert G\vert&if\;a_i=E\\0&otherwise\end{cases}\;\;\;\;\;\;\;\;36$

Question

Show that the regular representation is reducible.

Answer

If the regular representation is reducible, the LHS of eq28 must be greater than $\vert G\vert$ . Applying the LHS of eq28 to the regular representation and using eq36,

$\sum_{\alpha}n_{\alpha}\vert\chi_R(\alpha)\vert^2=n_1\vert\chi_R(1)\vert^2+n_2\vert\chi_R(2)\vert^2+\cdots =\vert G\vert^2$

For non-single element groups, $\vert G\vert>1$ and hence $\vert G\vert^2>\vert G\vert$ .

Finally, we shall prove that $\sum_kd_k^2=\vert G\vert$ . From eq25 and eq27, we have

$\chi_R(\alpha)=\sum_kc_k\chi_k(\alpha)\;\;\;\;\;\;\;\;37$

$c_k=\frac{1}{\vert G\vert}\sum__{\alpha}n_{\alpha}\chi_k^*(\alpha)\chi_R(\alpha)\;\;\;\;\;\;\;\;38$

where $\chi_R(\alpha)$ and $\chi_k(\alpha)$ are the characters for the $\alpha$ -th class of the regular representation and irreducible representation respectively.

Expanding eq38 and using eq36,

${c_k=\frac{1}{\vert G\vert}\sum__{\alpha}n_{\alpha}\chi_k^*(\alpha)\chi_R(\alpha)=\frac{1}{\vert G\vert}\[n_E\chi_k^*(E)\chi_R(E)+\cdots\]=\frac{1}{\vert G\vert}(1)d_k\vert G\vert=d_k}\;\;\;\;\;\;\;\;39$

Question

Why is $\chi_k^*(E)=d_k$ ?

Answer

The regular representation for $E$ is a reducible representation that is the direct sum of irreducible representations under the class $E$ . Each of these constituent irreducible representations is in general a $d_k\times d_k$ matrix with trace of $d_k$ .

Eq39 therefore states that each constituent irreducible representation occurs in the regular representation a number of times that is equal to the dimension of the corresponding irreducible representation. For example, with reference to the $C_{3v}$ table below, $\Gamma_R(E)=\Gamma_1(E)\oplus\Gamma_2(E)\oplus2\Gamma_3(E)$ , where $c_1=d_1=1$ , $c_2=d_2=1$ and $c_3=d_3=2$ . In other words, each of the irreducible representations $\Gamma_1$ and $\Gamma_2$ appears once in the $6\times 6$ regular representation matrix element $\Gamma_R(E)$ , while the irreducible representation $\Gamma_3$ appears twice.