Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it.
In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple representations in that sense.
[1] The following are equivalent:[2] The equivalence of the above conditions can be proved based on the following lemma, which is of independent interest: Lemma[3] — Let p:V → W be a surjective equivariant map between representations.
are subrepresentations; i.e., we can assume the direct sum is internal.
Shrinking W to a (nonzero) cyclic subrepresentation we can assume it is finitely generated.
to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation
:[5] The implication is a direct generalization of a basic fact in linear algebra that a basis can be extracted from a spanning set of a vector space.
That is we can prove the following slightly more precise statement: As in the proof of the lemma, we can find a maximal direct sum
[6] Hence, every finite-dimensional continuous complex representation of G is semisimple.
[7] For a finite group, this is a special case of Maschke's theorem, which says a finite-dimensional representation of a finite group G over a field k with characteristic not dividing the order of G is semisimple.
[10] Given a linear endomorphism T of a vector space V, V is semisimple as a representation of T (i.e., T is a semisimple operator) if and only if the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials.
[11] Given a finite-dimensional representation V, the Jordan–Hölder theorem says there is a filtration by subrepresentations:
in a natural way and makes V a representation of G. If W is a subrepresentation of V that has dimension 1, then a simple calculation shows that it must be spanned by the vector
That is, there are exactly three G-subrepresentations of V; in particular, V is not semisimple (as a unique one-dimensional subrepresentation does not admit a complementary representation).
[15] However, for a finite-dimensional semisimple representation V over an algebraically closed field, the numbers of simple representations up to isomorphism appearing in the decomposition of V (1) are unique and (2) completely determine the representation up to isomorphism;[16] this is a consequence of Schur's lemma in the following way.
are independent of chosen decompositions; they are the multiplicities of simple representations
is a unitary representation with the inner product given by the averaging argument, the Schur orthogonality relations say:[19] the irreducible characters (characters of simple representations) of G are an orthonormal subset of the space of complex-valued functions on G and thus
By definition, given a simple representation S, the isotypic component of type S of a representation V is the sum of all subrepresentations of V that are isomorphic to S;[15] note the component is also isomorphic to the direct sum of some choice of subrepresentations isomorphic to S (so the component is unique, while the summands are not necessary so).
is the set of isomorphism classes of simple representations of G and
be the space of homogeneous degree-three polynomials over the complex numbers in variables
This is a finite-dimensional complex representation of a finite group, and so is semisimple.
The most basic case of this is the Peter–Weyl theorem, which decomposes the left (or right) regular representation of a compact group into the Hilbert-space completion of the direct sum of all simple unitary representations.
= the Hilbert space of (classes of) square-integrable functions on a compact group G: where
means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations
is simply the group algebra of G and also the completion is vacuous.
[22] This is one of standard facts in the representation theory of a finite group (and is much easier to prove).
, the theorem exactly amounts to the classical Fourier analysis.
[23] In quantum mechanics and particle physics, the angular momentum of an object can be described by complex representations of the rotation group SO(3), all of which are semisimple.
[24] Due to connection between SO(3) and SU(2), the non-relativistic spin of an elementary particle is described by complex representations of SU(2) and the relativistic spin is described by complex representations of SL2(C), all of which are semisimple.
[24] In angular momentum coupling, Clebsch–Gordan coefficients arise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.