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C. S. G. Phillips coauthored reference tables connecting molecular symmetry with group theory. Their character tables organize how coordinates, rotations and other molecular quantities transform under a point group, supporting assignments in molecular spectroscopy.^[ref]

M. S. Child coauthored reference tables for applying group theory to molecules. A character table makes the symmetry behaviour of a vibration or electronic function comparable with the irreducible representations of the relevant molecular point group.^[ref]

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]

The highest expected orbital degeneracy is five. Ideal buckminsterfullerene has Ih molecular point group symmetry. Its irreducible orbital representations include five-dimensional $H_g$ and $H_u$ representations; no single-valued orbital irrep has higher dimension.

This counts distinct spatial molecular orbitals. It does not multiply by electron spin, and it excludes accidental coincidences between unrelated irreps. In a real environment, symmetry lowering can split these degeneracies.^[ref]