Semisimple representation

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 ${\displaystyle V=W\oplus W'}$.

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

Proof of the lemma: Write ${\displaystyle V=\bigoplus _{i\in I}V_{i}}$ where ${\displaystyle V_{i}}$ are simple representations. Without loss of generality, we can assume ${\displaystyle V_{i}}$ are subrepresentations; i.e., we can assume the direct sum is internal. Now, consider the family of all possible direct sums ${\displaystyle V_{J}:=\bigoplus _{i\in J}V_{i}\subset V}$ with various subsets ${\displaystyle J\subset I}$. Put the partial ordering on it by saying the direct sum over K is less than the direct sum over J if ${\displaystyle K\subset J}$. By Zorn's lemma, we can find a maximal ${\displaystyle J\subset I}$ such that ${\displaystyle \operatorname {ker} p\cap V_{J}=0}$. We claim that ${\displaystyle V=\operatorname {ker} p\oplus V_{J}}$. By definition, ${\displaystyle \operatorname {ker} p\cap V_{J}=0}$ so we only need to show that ${\displaystyle V=\operatorname {ker} p+V_{J}}$. If ${\displaystyle \operatorname {ker} p+V_{J}}$ is a proper subrepresentatiom of ${\displaystyle V}$ then there exists ${\displaystyle k\in I-J}$ such that ${\displaystyle V_{k}\not \subset \operatorname {ker} p+V_{J}}$. Since ${\displaystyle V_{k}}$ is simple (irreducible), ${\displaystyle V_{k}\cap (\operatorname {ker} p+V_{J})=0}$. This contradicts the maximality of ${\displaystyle J}$, so ${\displaystyle V=\operatorname {ker} p\oplus V_{J}}$ as claimed. Hence, ${\displaystyle W\simeq V/\operatorname {ker} p\simeq V_{J}\to V}$ is a section of p. ${\displaystyle \square }$

Note that we cannot take ${\displaystyle J}$ to the set of ${\displaystyle i}$ such that ${\displaystyle \ker(p)\cap V_{i}=0}$. The reason is that it can happen, and frequently does, that ${\displaystyle X}$ is a subspace of ${\displaystyle Y\oplus Z}$ and yet ${\displaystyle X\cap Y=0=X\cap Z}$. For example, take ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$ to be three distinct lines through the origin in ${\displaystyle \mathbb {R} ^{2}}$. For an explicit counterexample, let ${\displaystyle A=\operatorname {Mat} _{2}(F)}$ be the algebra of ${\displaystyle 2\times 2}$ matrices and set ${\displaystyle V=A}$, the regular representation of ${\displaystyle A}$. Set ${\displaystyle V_{1}={\Bigl \{}{\begin{pmatrix}a&0\\b&0\end{pmatrix}}{\Bigr \}}}$ and ${\displaystyle V_{2}={\Bigl \{}{\begin{pmatrix}0&c\\0&d\end{pmatrix}}{\Bigr \}}}$ and set ${\displaystyle W={\Bigl \{}{\begin{pmatrix}c&c\\d&d\end{pmatrix}}{\Bigr \}}}$. Then ${\displaystyle V_{1}}$, ${\displaystyle V_{2}}$ and ${\displaystyle W}$ are all irreducible ${\displaystyle A}$-modules and ${\displaystyle V=V_{1}\oplus V_{2}}$. Let ${\displaystyle p:V\to V/W}$ be the natural surjection. Then ${\displaystyle \operatorname {ker} p=W\neq 0}$ and ${\displaystyle V_{1}\cap \operatorname {ker} p=0=V_{2}\cap \operatorname {ker} p}$. In this case, ${\displaystyle W\simeq V_{1}\simeq V_{2}}$ but ${\displaystyle V\neq \operatorname {ker} p\oplus V_{1}\oplus V_{2}}$ because this sum is not direct.

Proof of equivalences[4] ${\displaystyle 1.\Rightarrow 3.}$: Take p to be the natural surjection ${\displaystyle V\to V/W}$. Since V is semisimple, p splits and so, through a section, ${\displaystyle V/W}$ is isomorphic to a subrepretation that is complementary to W.

${\displaystyle 3.\Rightarrow 2.}$: 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., ${\displaystyle V=U\oplus U'}$ for some ${\displaystyle U'}$. By modular law, it implies ${\displaystyle W=U\oplus (W\cap U')}$. Then ${\displaystyle (W\cap U')\simeq W/U}$ is a simple subrepresentation of W ("simple" because of maximality). This establishes the observation. Now, take ${\displaystyle W}$ to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation ${\displaystyle W'}$. If ${\displaystyle W'\neq 0}$, then, by the early observation, ${\displaystyle W'}$ contains a simple subrepresentation and so ${\displaystyle W\cap W'\neq 0}$, a nonsense. Hence, ${\displaystyle W'=0}$.

${\displaystyle 2.\Rightarrow 1.}$:[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 ${\displaystyle V=\sum _{i\in I}V_{i}}$ is a sum of simple subrepresentations, a semisimple decomposition ${\displaystyle V=\bigoplus _{i\in I'}V_{i}}$, some subset ${\displaystyle I'\subset I}$, can be extracted from the sum.

As in the proof of the lemma, we can find a maximal direct sum ${\displaystyle W}$ that consists of some ${\displaystyle V_{i}}$’s. Now, for each i in I, by simplicity, either ${\displaystyle V_{i}\subset W}$ or ${\displaystyle V_{i}\cap W=0}$. In the second case, the direct sum ${\displaystyle W\oplus V_{i}}$ is a contradiction to the maximality of W. Hence, ${\displaystyle V_{i}\subset W}$. ${\displaystyle \square }$

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]

${\displaystyle V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}}$

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

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

where ${\displaystyle \operatorname {Hom} _{\text{equiv}}}$ refers to the equivariant linear maps. Also, each ${\displaystyle m_{i}}$ is unchanged if ${\displaystyle V_{i}}$ is replaced by another simple representation isomorphic to ${\displaystyle V_{i}}$. Thus, the integers ${\displaystyle m_{i}}$ are independent of chosen decompositions; they are the multiplicities of simple representations ${\displaystyle V_{i}}$, up to isomorphisms, in V.[17]

In general, given a finite-dimensional representation ${\displaystyle \pi :G\to GL(V)}$ of a group G over a field k, the composition ${\displaystyle \chi _{V}:G{\overset {\pi }{\to }}GL(V){\overset {\text{tr}}{\to }}k}$ is called the character of ${\displaystyle (\pi ,V)}$.[18] When ${\displaystyle (\pi ,V)}$ is semisimple with the decomposition ${\displaystyle V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}}$ as above, the trace ${\displaystyle \operatorname {tr} (\pi (g))}$ is the sum of the traces of ${\displaystyle \pi (g):V_{i}\to V_{i}}$ with multiplicities and thus, as functions on G,

${\displaystyle \chi _{V}=\sum _{i}m_{i}\chi _{V_{i}}}$

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

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]

${\displaystyle V=\bigoplus _{\lambda \in {\widehat {G}}}V^{\lambda }}$

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

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 ${\displaystyle W=L^{2}(G)}$ = the Hilbert space of (classes of) square-integrable functions on a compact group G:

${\displaystyle W\simeq {\widehat {\bigoplus _{[(\pi ,V)]}}}V^{\oplus \dim V}}$

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

${\displaystyle \mathbb {C} [G]=\bigoplus _{[(\pi ,V)]}V^{\oplus \dim V}.}$

That is, each simple representation of G appears in the regular representation with multiplicity the dimension of the representation.[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 ${\displaystyle S^{1}}$, the theorem exactly amounts to the classical Fourier analysis.[23]

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]