An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces. For example, the orthogonal group O(n) has an irreducible representation on R^n. Any representation of a finite or semisimple Lie group breaks up into a direct sum of irreducible representations. But in general, this is not the case, e.g., (R, + ) has a representation on R^2 by ϕ(a) = [1 | a 0 | 1], i.e., ϕ(a)(x, y) = (x + a y, y). But the subspace y = 0 is fixed, hence ϕ is not irreducible, but there is no complementary invariant subspace.
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