# 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 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:

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 $V=W\oplus W'$ .

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

Lemma  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 because if $v\in W^{\bot }$ and $g\in G$ , then $\langle \pi (g)v,w\rangle =\langle v,\pi (g^{-1})w\rangle =0$ for any w in W since W is G-invariant, and so $\pi (g)v\in W^{\bot }$ .

For example, given a continuous finite-dimensional complex representation $\pi :G\to GL(V)$ of a finite group or a compact group G, by the averaging argument, one can define an inner product $\langle ,\rangle$ on V that is G-invariant: i.e., $\langle \pi (g)v,\pi (g)w\rangle =\langle v,w\rangle$ , which is to say $\pi (g)$ is a unitary operator and so $\pi$ 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.

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

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.

A representation of a unipotent group is generally not semisimple. Take $G$ to be the group consisting of real matrices ${\begin{bmatrix}1&a\\&1\end{bmatrix}}$ ; it acts $V=\mathbb {R} ^{2}$ 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 ${\begin{bmatrix}1\\0\end{bmatrix}}$ . 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):

$V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}$ where $V_{i}$ are simple representations, mutually non-isomorphic to one another, and $m_{i}$ are positive integers. By Schur's lemma,

$m_{i}=\dim \operatorname {Hom} _{\text{equiv}}(V_{i},V)=\dim \operatorname {Hom} _{\text{equiv}}(V,V_{i})$ ,

where $\operatorname {Hom} _{\text{equiv}}$ refers to the equivariant linear maps. Also, each $m_{i}$ is unchanged if $V_{i}$ is replaced by another simple representation isomorphic to $V_{i}$ . Thus, the integers $m_{i}$ are independent of chosen decompositions; they are the multiplicities of simple representations $V_{i}$ , up to isomorphisms, in V.

In general, given a finite-dimensional representation $\pi :G\to GL(V)$ of a group G over a field k, the composition $\chi _{V}:G{\overset {\pi }{\to }}GL(V){\overset {\text{tr}}{\to }}k$ is called the character of $(\pi ,V)$ . When $(\pi ,V)$ is semisimple with the decomposition $V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}$ as above, the trace $\operatorname {tr} (\pi (g))$ is the sum of the traces of $\pi (g):V_{i}\to V_{i}$ with multiplicities and thus, as functions on G,

$\chi _{V}=\sum _{i}m_{i}\chi _{V_{i}}$ where $\chi _{V_{i}}$ are the characters of $V_{i}$ . When G is a finite group or more generally a compact group and $V$ 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 $m_{i}=\langle \chi _{V},\chi _{V_{i}}\rangle$ .

## 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:

$V=\bigoplus _{\lambda \in {\widehat {G}}}V^{\lambda }$ where ${\widehat {G}}$ is the set of isomorphism classes of simple representations of V and $V^{\lambda }$ is the isotypic component of V of type S for some $S\in \lambda$ .

## 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: given $W=L^{2}(G)$ = the Hilbert space of (classes of) square-integrable functions on a compact group G, there is a natural decomposition:

$W\simeq {\widehat {\bigoplus _{[(\pi ,V)]}}}V^{\oplus \dim V}$ where ${\widehat {\bigoplus }}$ means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations $(\pi ,V)$ 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 $W=\mathbb {C} [G]$ is simply the group algebra of G and also the completion is vacuous. Thus, the theorem simply says that

$\mathbb {C} [G]=\bigoplus _{[(\pi ,V)]}V^{\oplus \dim V}.$ 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 $S^{1}$ , the theorem exactly amounts to the classical Fourier analysis.

## Sources

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