Now for a third mathematical definition of irreducible, this time from

Representation Theory:

**Definition** Suppose (ρ,V) is a representation of some group G. If there is **no** proper non-trivial subspace W of V such that ρ(G)W is contained in W, then the representation is said to be irreducible.

Come to think of it, Representation Theory is repsonsible for a great many overloaded definitions;

simple is also an equivalent condition to irreducible, then there's

complete,

regular and

characteristic which spring instantly to mind which also have definitions elsewhere in mathematics.

What is nice about the idea of irreducibility is that for

finite and indeed

compact groups,

*every* representation of the group can be split up into a

direct sum of irreducible ones, where the number of such irreducible representations is equal to the number of

conjugacy classes of the group.