completely reducible

A completely reducible representation is one that falls apart cleanly into irreducible blocks, like a block-diagonal matrix with simple bricks on the diagonal and nothing tangling them together. The word 'completely' is the key: not only does the representation contain irreducible pieces, but the whole thing is exactly the direct sum of such pieces, with no leftover gluing.

Formally, a representation V of G is completely reducible (or semisimple) if it is a direct sum V = V_1 + V_2 + ... + V_r of irreducible subrepresentations. An equivalent and often more useful criterion is that every subrepresentation W of V has a G-invariant complement: a subrepresentation W' with V = W + W' as an internal direct sum. This complement condition is what fails for general modules and makes semisimplicity special.

The distinction between 'reducible' and 'completely reducible' is real. A representation can have a proper invariant subspace yet refuse to split. The standard example is the group of upper-triangular unipotent matrices [1, t; 0, 1] for t in k acting on k^2: the line spanned by (1, 0) is invariant but has no invariant complement, so this 2-dimensional representation is reducible but not completely reducible.