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 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].

Equivalent characterizationsEdit

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]

  1. V is semisimple as a representation.
  2. V is a sum of simple subrepresentations.
  3. 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:VW be a surjective equivariant map between representations. If V is semisimple, then p splits; i.e., it admits a section.

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[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 we can prove the following slightly more precise statement:

  • 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-examplesEdit

Unitary representationsEdit

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]

Representations of semisimple Lie algebrasEdit

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]

Separable minimal polynomialsEdit

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]

Associated semisimple representationEdit

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.[12]

Unipotent group non-exampleEdit

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).[13]

Semisimple decomposition and multiplicityEdit

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.[14] The isotypic decomposition, on the other hand, is an example of a unique decomposition.[15]

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;[16] 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):[16]


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.[17]

In general, given a finite-dimensional representation   of a group G over a field k, the composition   is called the character of  .[18] 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:[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  .

Isotypic decompositionEdit

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;[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).

Then the isotypic decomposition of a semisimple representation V is the (unique) direct sum decomposition:[15][20]


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,[21] 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.[22] 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.[23]

Applications to physicsEdit

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.[25]


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



  1. ^ a b Procesi 2007, Ch. 6, § 1.1, Definition 1 (ii).
  2. ^ Procesi 2007, Ch. 6, § 2.1.
  3. ^ Anderson & Fuller 1992, Proposition 9.4.
  4. ^ Anderson & Fuller 1992, Theorem 9.6.
  5. ^ Anderson & Fuller 1992, Lemma 9.2.
  6. ^ a b Fulton & Harris 1991, § 9.3. A
  7. ^ Hall 2015, Theorem 4.28
  8. ^ Fulton & Harris 1991, Corollary 1.6.
  9. ^ Serre 1977, Theorem 2.
  10. ^ Hall 2015 Theorem 10.9
  11. ^ Jacobson 1989, § 3.5. Exercise 4.
  12. ^ Artin 1999, Ch. V, § 14.
  13. ^ Fulton & Harris 1991, just after Corollary 1.6.
  14. ^ Serre 1977, § 1.4. remark
  15. ^ a b c Procesi 2007, Ch. 6, § 2.3.
  16. ^ a b Fulton & Harris 1991, Proposition 1.8.
  17. ^ Fulton & Harris 1991, § 2.3.
  18. ^ Fulton & Harris 1991, § 2.1. Definition
  19. ^ Serre 1977, § 2.3. Theorem 3 and § 4.3.
  20. ^ Serre 1977, § 2.6. Theorem 8 (i)
  21. ^ Procesi 2007, Ch. 8, Theorem 3.2.
  22. ^ Serre 1977, § 2.4. Corollary 1 to Proposition 5
  23. ^ Procesi 2007, Ch. 8, § 3.3.
  24. ^ a b Hall, Brian C. (2013). "Angular Momentum and Spin". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. Springer. pp. 367–392. ISBN 978-1461471158.
  25. ^ Klimyk, A. U.; Gavrilik, A. M. (1979). "Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups". Journal of Mathematical Physics. 20 (1624): 1624–1642. Bibcode:1979JMP....20.1624K. doi:10.1063/1.524268.


  • Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2nd ed.), New York, NY: 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).
  • Artin, Michael (1999). "Noncommutative Rings" (PDF).
  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 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. Vol. 222 (2nd ed.). Springer. ISBN 978-3319134666.
  • Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5
  • Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402..
  • Serre, Jean-Pierre (1977-09-01). Linear Representations of Finite Groups. Graduate Texts in Mathematics, 42. New York–Heidelberg: Springer-Verlag. ISBN 978-0-387-90190-9. MR 0450380. Zbl 0355.20006.