Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / b / iii / Solution

Taking the matrix trace of those matrices givesThe one-dimensional $p_z$ subspace has characters $(1,-1,1,-1,1)$, identifying $B_2$. The pair $(p_x,p_y)$ has characters $(2,0,-2,0,0)$, identifying $E$. ThereforeThe dimensions add to three, as required. The $p_x,p_y$ pair must be retained together: a ninety-degree improper rotation mixes them, so either one alone is not a complete symmetry basis.^[ref]