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). 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 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.
The following are equivalent:
The equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:
Examples and non-examples.
Unitary representations.
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 because if formula_2 and formula_3, then formula_4 for any "w" in "W" since "W" is "G"-invariant, and so formula_5.
For example, given a continuous finite-dimensional complex representation formula_6 of a finite group or a compact group "G", by the averaging argument, one can define an inner product formula_7 on "V" that is "G"-invariant: i.e., formula_8, which is to say formula_9 is a unitary operator and so formula_10 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" (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.
Associated semisimple representation.
Given a finite-dimensional representation "V", the Jordan–Hölder theorem says there is a filtration by subrepresentations: formula_11 such that each successive quotient formula_12 is a simple representation. Then the associated vector space formula_13 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 formula_14 to be the group consisting of real matrices formula_15; it acts on formula_16 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 formula_17. 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).
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 isomorphisms appearing in the decomposition of "V" (1) are unique and (2) completely determine the representation up to isomorphisms; 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):
where formula_19 are simple representations, mutually non-isomorphic to one another, and formula_20 are positive integers. By Schur's lemma,
where formula_22 refers to the equivariant linear maps. Also, each formula_20 is unchanged if formula_19 is replaced by another simple representation isomorphic to formula_19. Thus, the integers formula_20 are independent of chosen decompositions; they are the "multiplicities" of simple representations formula_19, up to isomorphisms, in "V".
In general, given a finite-dimensional representation formula_6 of a group "G" over a field "k", the composition formula_29 is called the character of formula_30. When formula_30 is semisimple with the decomposition formula_18 as above, the trace formula_33 is the sum of the traces of formula_34 with multiplicities and thus, as functions on "G",
where formula_36 are the characters of formula_19. When "G" is a finite group or more generally a compact group and formula_38 is a unitary representation with the inner product given by the averaging argument, the Schur orthogonality relations say: the irreducible characters (characters of simple representations) of "G" are an orthonormal subset of the space of complex-valued functions on "G" and thus formula_39.
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" (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:
where formula_41 is the set of isomorphism classes of simple representations of "V" and formula_42 is the isotypic component of "V" of type "S" for some formula_43.
Example.
Let formula_38 be the space of homogeneous degree-three polynomials over the complex numbers in variables formula_45. Then formula_46 acts on formula_38 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 formula_46. In particular, formula_38 contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation formula_50 of formula_46. For example, the span of formula_52 and formula_53 is isomorphic to formula_50. This can more easily be seen by writing this two-dimensional subspace as 
Another copy of formula_50 can be written in a similar form:
So can the third:
Then formula_59 is the isotypic component of type formula_50 in formula_38. 
Completion.
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, there is a natural decomposition for formula_62 = the Hilbert space of (classes of) square-integrable functions on a compact group "G":
where formula_64 means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations formula_30 of "G". 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 formula_66 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. 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 formula_68, the theorem exactly amounts to the classical Fourier analysis.
Applications to physics.
In quantum mechanics and particle physics, the angular momentum of an object can be described by , all of which are semisimple. 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. 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.