I am reading a note on representation theory; a Dirac Character is defined as:
$$\Omega_{a}=\sum_{g \in \mathcal{C}_{a}} g$$
Then the author argues that $\Omega_{a}$ commutes with all group elements, since $g^{-1} \Omega_{a} g$ simply reorders the elements. This causes confusion…
A conjugacy class is defined as $\left\{f g f^{-1}: f \in G\right\}$ for a fixed $f$, so a Dirac Character should only commute with this particular $f$, but why does it commute with all the group elements?
Thanks in Advance!
Best Answer
By definition, $C_a = \{gag^{-1}:g \in G\}$, so given any $h \in G$, $$h \Omega_a h^{-1} = \sum_{g \in C_a} hgh^{-1}$$ But $g$ is obtained by conjugating $a$, so $hgh^{-1}$ is also some conjugation of $a$. It is then clear that we are simply permuting the sum, so $$h \Omega_a h^{-1} = \Omega_a$$