Semisimple representation

In mathematics, specifically in representation theory, a semisimple representation is a linear representation of a group or an algebra that is a direct sum of simple 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 algebra k.

Equivalent characterizations

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 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:

V is semisimple as a representation.
V is a sum of simple subrepresentations.
Each subrepresentation W of V admits a complementary representation: a subrepresentation W such that.

The equivalence of the above conditions can be proved based on the following 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,. 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 2-by-2 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 subrepresentation that is complementary to W.
of W. 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,.

When is a sum of simple subrepresentations, a semisimple decomposition, some subset, can be extracted from the sum.

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,.

Examples and non-examples

Unitary representations

A finite-dimensional unitary representation 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.

Representations of semisimple Lie algebras

By Weyl's theorem on complete reducibility, every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is semisimple.

Separable minimal polynomials

Given a linear endomorphism T of a vector space V, V is semisimple as a representation of T if and only if the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials.

Associated semisimple representation

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.

Unipotent group non-example

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.

Semisimple decomposition and multiplicity

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 isomorphism appearing in the decomposition of V are unique and completely determine the representation up to isomorphism; 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 :
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 isomorphism, 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 of G are an orthonormal subset of the space of complex-valued functions on G and thus.

Isotypic decomposition

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.
Then the isotypic decomposition of a semisimple representation V is the direct sum decomposition:
where is the set of isomorphism classes of simple representations of G and is the isotypic component of V of type S for some.

Isotypic component

The isotypic component of weight of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight.

Definition

A finite-dimensional module of a reductive Lie algebra can be decomposed into irreducible submodules
Each finite-dimensional irreducible representation of is uniquely identified by its highest weight
In the decomposition of, a certain isomorphism class might appear more than once, hence

This defines the isotypic component of weight of : where is maximal.

Example

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.