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]

  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-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

  1. Procesi, Ch. 6, § 1.1, Definition 1 (ii).
  2. Procesi, Ch. 6, § 2.1.
  3. Anderson & Fuller, Proposition 9.4.
  4. Anderson & Fuller, Theorem 9.6.
  5. Anderson & Fuller, Lemma 9.2.
  6. Fulton & Harris, § 9.3. A
  7. Hall 2015, Theorem 4.28
  8. Fulton & Harris, Corollary 1.6.
  9. Serre, Theorem 2.
  10. Hall 2015 Theorem 10.9
  11. Jacobson, § 3.5. Exercise 4.
  12. Fulton & Harris, just after Corollary 1.6.
  13. Serre, § 1.4. remark
  14. Procesi, Ch. 6, § 2.3.
  15. Fulton & Harris, Proposition 1.8.
  16. Fulton & Harris, § 2.3.
  17. Fulton & Harris, § 2.1. Definition
  18. Serre, § 2.3. Theorem 3 and § 4.3.
  19. Serre, § 2.6. Theorem 8 (i)
  20. Procesi, Ch. 8, Theorem 3.2.
  21. Serre, § 2.4. Corollary 1 to Proposition 5
  22. Procesi, Ch. 8, § 3.3.
  1. To be precise, the theorem concerns the regular representation of and the above statement is a corollary.

Sources

  • Anderson, Frank W.; Fuller, Kent R. (1992), 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). 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.
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