# Semisimple representation

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).[1] It is an example of the general mathematical notion of semisimplicity.

Many representations that appear in applications 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*].

## 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 (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]

*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 equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:

**Lemma[3]** — Let *p*:*V* → *W* be a surjective equivariant map between representations. If *V* is semisimple, then *p* splits; i.e., it admits a section.

**Proof of 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. By simplicity, either or . Thus, where is such that for each , . Then is a section of *p*.

**Proof of equivalences[4]**

: 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, .

:[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 the statement we can show is: when is a sum of simple subrepresentations, a semisimple decomposition , some subset , can be extracted from the sum. 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 . Zorn's lemma clearly applies to it and gives us a maximal direct sum *W*. 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

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[6] 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.[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.[8][9]

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

A representation of a unipotent group is generally not semisimple. Take to be the group consisting of real matrices ; it acts on 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).[12]

## 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.[13] The isotypic decomposition, on the other hand, is an example of a unique decomposition.[14]

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;[15] 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):[15]

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*.[16]

In general, given a finite-dimensional representation of a group *G* over a field *k*, the composition is called the character of .[17] 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:[18] the irreducible characters (characters of simple representations) 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*;[14] 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).

Then the isotypic decomposition of a semisimple representation *V* is the (unique) direct sum decomposition:[14][19]

where is the set of isomorphism classes of simple representations of *V* and is the isotypic component of *V* of type *S* for some .

## The completion of a semisimple representation

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. Precisely, it says:[20] given = the Hilbert space of (classes of) square-integrable functions on a compact group *G*, there is a natural decomposition:

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.[21] This is one of standard facts in the representation theory of a finite group (and is much easier to prove).

When the group *G* is the circle group , the theorem exactly amounts to the classical Fourier analysis.[22]

## References

- Procesi, Ch. 6, § 1.1, Definition 1 (ii).
- Procesi, Ch. 6, § 2.1.
- Anderson & Fuller, Proposition 9.4.
- Anderson & Fuller, Theorem 9.6.
- Anderson & Fuller, Lemma 9.2.
- Fulton & Harris, § 9.3. A
- Hall 2015, Theorem 4.28
- Fulton & Harris, Corollary 1.6.
- Serre, Theorem 2.
- Hall 2015 Theorem 10.9
- Jacobson, § 3.5. Exercise 4.
- Fulton & Harris, just after Corollary 1.6.
- Serre, § 1.4. remark
- Procesi, Ch. 6, § 2.3.
- Fulton & Harris, Proposition 1.8.
- Fulton & Harris, § 2.3.
- Fulton & Harris, § 2.1. Definition
- Serre, § 2.3. Theorem 3 and § 4.3.
- Serre, § 2.6. Theorem 8 (i)
- Procesi, Ch. 8, Theorem 3.2.
- Serre, § 2.4. Corollary 1 to Proposition 5
- Procesi, Ch. 8, § 3.3.

- To be precise, the theorem concerns the regular representation of and the above statement is a corollary.

## Sources

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*Rings and categories of modules*, Graduate Texts in Mathematics,**13**(2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487; NB: this reference, nominally, considers a semisimple module over a ring not over a group but this is not a material difference (the abstract part of the discussion goes through for groups as well). - Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics.**129**. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. - Hall, Brian C. (2015).
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*. Graduate Texts in Mathematics.**222**(2nd ed.). Springer. ISBN 978-3319134666. - Jacobson, Nathan (1989),
*Basic algebra II*(2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5 - Claudio Procesi (2007)
*Lie Groups: an approach through invariants and representation*, Springer, ISBN 9780387260402. - Serre, Jean-Pierre (1977-09-01).
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