Past exam of the groups and representations course of the University of Oxford / 2021 / 1 / b / Solution

Write the two Pauli matrices asThey satisfy $A^2=B^2=I$, $BA=-AB$ and $C^2=-I$. Moving every $A$ to the left of every $B$ reduces each word to one of eight distinct matrices. ConsequentlyTaking $r=C$, $s=A$ gives $r^4=s^2=I$ and $srs=r^{-1}$, identifying the group with the dihedral group of a square.

The elements $\pm I$ are central. Conjugating $A$ by $B$ changes its sign, and similarly for $B$ and $C$. No conjugation mixes these pairs: modulo the central subgroup $\{\pm I\}$ the quotient group is the Klein four-group and hence Abelian. Thus the conjugacy classes are preciselyThere are five complex irreducible representations, one for each conjugacy class. In a one-dimensional representation, the images of $A,B$ are signs $s,t\in \{1,-1\}$. Since they commute, $-I$ maps to $1$, and all four choices of $s,t$ satisfy the relations. These are four inequivalent one-dimensional representations. The sum of squares of irreducible representation dimensions leaves $8-4=4$ for the fifth, so the dimensions are