reducible representation

Suppose you want to know how a set of things — say all the C-H stretching motions of methane, or all the ligand sigma orbitals pointing at a metal — behaves under a molecule's symmetry. You start by treating the whole set as one big collection and ask, for each symmetry operation, how many members stay put. That collection of counts is a reducible representation: a composite symmetry description that mixes several fundamental types together and therefore can be broken down into simpler pieces.

Building one is mechanical. Pick a basis (the set of objects you care about), apply each symmetry operation, and for each operation write down the character — usually just how many basis objects are left unmoved (an object that swaps with another contributes zero; one that stays exactly in place contributes one, with sign and other rules for vectors). The resulting string of numbers, one per operation, is the reducible representation, often written with the Greek letter Gamma. By itself it is a jumble; the point is that it can always be expressed as a sum of the group's irreducible representations.

Turning that jumble into a clean sum of irreps is called reducing the representation, and a simple arithmetic formula (the reduction formula, using the characters and the number of operations in each class) does it. The result is the payoff of the whole subject: reduce the representation of all the atomic displacements and you find out how many vibrations there are of each symmetry type, and hence which are infrared- or Raman-active; reduce the representation of the ligand orbitals and you learn which symmetry-adapted combinations to draw on a molecular-orbital or ligand-field diagram. The reducible representation is the bridge from a real molecule to the predictions in the character table.