irreducible representation

Think of the irreducible representations of a point group as its primary colors: a small, fixed set of fundamental symmetry behaviors out of which every more complicated behavior can be built, but which cannot themselves be broken into anything simpler. Each one describes a distinct way that something — an orbital, a vibration, a vector — can respond to the molecule's symmetry operations. They are the rows of the character table, and they carry the short Mulliken labels you see everywhere in inorganic chemistry: a1, e, t2g, and the rest.

Each irreducible representation (often abbreviated "irrep") is summarized by its set of characters — one number per class of operations — telling you how that symmetry type transforms. The Mulliken labels encode this: a or b means a one-dimensional (non-degenerate) type, with a symmetric and b antisymmetric under the principal rotation; e means a doubly-degenerate pair that mix together; t means a triply-degenerate trio. Subscripts add detail: 1 or 2 for behavior under a secondary axis or plane, and g (gerade) or u (ungerade) for symmetric or antisymmetric under inversion in centrosymmetric groups. So "t2g" reads as a triply-degenerate set, of type 2, that is symmetric under inversion — exactly the label for one of the split d-orbital sets in an octahedral complex.

Why do these labels run through all of bonding and spectroscopy? Because every orbital, every molecular vibration, and every electronic state must belong to one of a molecule's irreducible representations — symmetry leaves no other option. Knowing which irrep a thing belongs to immediately tells you what it can interact with: only orbitals of the same irrep can combine to bond, only vibrations of the right irrep are infrared- or Raman-active, and electronic transitions are allowed or forbidden depending on how the irreps multiply together. The irreducible representations are, quite literally, the alphabet of symmetry-based chemistry.