In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity.
Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group G over a field k is a semisimple module over the group ring k[G].
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.
The following are equivalent:
The equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:
Proof of the lemma: Write where are simple representations. Without loss of generality, we can assume are subrepresentations; i.e., we can assume the direct sum is internal. Now, consider the family of all possible direct sums with various subsets . Put the partial ordering on it by saying the direct sum over K is less than the direct sum over J if . By Zorn's lemma, we can find a maximal such that . We claim that . By definition, so we only need to show that . If is a proper subrepresentatiom of then there exists such that . Since is simple (irreducible), . This contradicts the maximality of , so as claimed. Hence, is a section of p.
Note that we cannot take to the set of such that . The reason is that it can happen, and frequently does, that is a subspace of and yet . For example, take , and to be three distinct lines through the origin in . For an explicit counterexample, let be the algebra of matrices and set , the regular representation of . Set and and set . Then , and are all irreducible -modules and . Let be the natural surjection. Then and . In this case, but because this sum is not direct.
Proof of equivalences : Take p to be the natural surjection . Since V is semisimple, p splits and so, through a section, is isomorphic to a subrepretation that is complementary to W.
: We shall first observe that every nonzero subrepresentation W has a simple subrepresentation. Shrinking W to a (nonzero) cyclic subrepresentation we can assume it is finitely generated. Then it has a maximal subrepresentation U. By the condition 3., for some . By modular law, it implies . Then is a simple subrepresentation of W ("simple" because of maximality). This establishes the observation. Now, take to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation . If , then, by the early observation, contains a simple subrepresentation and so , a nonsense. Hence, .
: 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 that consists of some ’s. Now, for each i in I, by simplicity, either or . In the second case, the direct sum is a contradiction to the maximality of W. Hence, .
A finite-dimensional unitary representation (i.e., a representation factoring through a unitary group) is a basic example of a semisimple representation. Such a representation is semisimple since if W is a subrepresentation, then the orthogonal complement to W is a complementary representation because if and , then for any w in W since W is G-invariant, and so .
For example, given a continuous finite-dimensional complex representation of a finite group or a compact group G, by the averaging argument, one can define an inner product on V that is G-invariant: i.e., , which is to say is a unitary operator and so is a unitary representation. Hence, every finite-dimensional continuous complex representation of G is semisimple. 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.
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.
Given a finite-dimensional representation V, the Jordan–Hölder theorem says there is a filtration by subrepresentations: such that each successive quotient is a simple representation. Then the associated vector space is a semisimple representation called an associated semisimple representation, which, up to an isomorphism, is uniquely determined by V.
A representation of a unipotent group is generally not semisimple. Take to be the group consisting of real matrices ; it acts on 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).
The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example, for a trivial representation, simple representations are one-dimensional vector spaces and thus a semisimple decomposition amounts to a choice of a basis of the representation vector space. The isotypic decomposition, on the other hand, is an example of a unique decomposition.
However, for a finite-dimensional semisimple representation V over an algebraically closed field, the numbers of simple representations up to isomorphisms appearing in the decomposition of V (1) are unique and (2) completely determine the representation up to isomorphisms; this is a consequence of Schur's lemma in the following way. Suppose a finite-dimensional semisimple representation V over an algebraically closed field is given: by definition, it is a direct sum of simple representations. By grouping together simple representations in the decomposition that are isomorphic to each other, up to an isomorphism, one finds a decomposition (not necessarily unique):
where are simple representations, mutually non-isomorphic to one another, and are positive integers. By Schur's lemma,
where refers to the equivariant linear maps. Also, each is unchanged if is replaced by another simple representation isomorphic to . Thus, the integers are independent of chosen decompositions; they are the multiplicities of simple representations , up to isomorphisms, in V.
In general, given a finite-dimensional representation of a group G over a field k, the composition is called the character of . When is semisimple with the decomposition as above, the trace is the sum of the traces of with multiplicities and thus, as functions on G,
where are the characters of . When G is a finite group or more generally a compact group and is a unitary representation with the inner product given by the averaging argument, the Schur orthogonality relations say: the irreducible characters (characters of simple representations) of G are an orthonormal subset of the space of complex-valued functions on G and thus .
There is a decomposition of a semisimple representation that is unique, called the isotypic decomposition of the representation. 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; 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).
where is the set of isomorphism classes of simple representations of G and is the isotypic component of V of type S for some .
Let be the space of homogeneous degree-three polynomials over the complex numbers in variables . Then acts on by permutation of the three variables. This is a finite-dimensional complex representation of a finite group, and so is semisimple. Therefore, this 10-dimensional representation can be broken up into three isotypic components, each corresponding to one of the three irreducible representations of . In particular, contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation of . For example, the span of and is isomorphic to . This can more easily be seen by writing this two-dimensional subspace as
Another copy of can be written in a similar form:
So can the third:
Then is the isotypic component of type in .
In Fourier analysis, one decomposes a (nice) function as the limit of the Fourier series of the function. In much the same way, a representation itself may not be semisimple but it may be the completion (in a suitable sense) of a semisimple representation. 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. As a corollary, there is a natural decomposition for = 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 of G.[note 1] Note here that every simple unitary representation (up to an isomorphism) appears in the sum with the multiplicity the dimension of the representation.
When the group G is a finite group, the vector space is simply the group algebra of G and also the completion is vacuous. Thus, the theorem simply says that
That is, each simple representation of G appears in the regular representation with multiplicity the dimension of the representation. This is one of standard facts in the representation theory of a finite group (and is much easier to prove).
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. 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. 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.