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.

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 V and   is the isotypic component of V of type S for some  .

CompletionEdit

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]

NotesEdit

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

ReferencesEdit

CitationsEdit

  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. 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. doi:10.1063/1.524268.

SourcesEdit