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.

The following are equivalent:{{harvnb|Procesi|2007|loc=Ch. 6, § 2.1.}}

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 $V\; =\; W\; \backslash oplus\; W\text{'}$.

The equivalence of the above conditions can be proved based on the following lemma, which is of independent interest:

{{math_theorem|name=Lemma{{harvnb|Anderson|Fuller|1992|loc=Proposition 9.4.}}|math_statement=Let p:V → W be a surjective equivariant map between representations. If V is semisimple, then p splits; i.e., it admits a section.}}

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

Note that we cannot take $J$ to the set of $i$ such that $\backslash ker(p)\; \backslash cap\; V\_i\; =\; 0$. The reason is that it can happen, and frequently does, that $X$ is a subspace of $Y\backslash oplus\; Z$ and yet $X\backslash cap\; Y\; =\; 0\; =\; X\backslash cap\; Z$. For example, take $X$, $Y$ and $Z$ to be three distinct lines through the origin in $\backslash mathbb\{R\}^2$. For an explicit counterexample, let $A\; =\; \backslash operatorname\{Mat\}\_2\; F$ be the algebra of 2-by-2 matrices and set $V=A$, the regular representation of $A$. Set and and set . Then $V\_1$, $V\_2$ and $W$ are all irreducible $A$-modules and $V\; =\; V\_1\backslash oplus\; V\_2$. Let $p:\; V\backslash to\; V/W$ be the natural surjection. Then $\backslash operatorname\{ker\}p\; =\; W\; \backslash ne\; 0$ and $V\_1\; \backslash cap\; \backslash operatorname\{ker\}p\; =\; 0\; =\; V\_2\; \backslash cap\; \backslash operatorname\{ker\}p$. In this case, $W\backslash simeq\; V\_1\backslash simeq\; V\_2$ but $V\; \backslash ne\; \backslash operatorname\{ker\}p\; \backslash oplus\; V\_1\backslash oplus\; V\_2$ because this sum is not direct.

Proof of equivalences{{harvnb|Anderson|Fuller|1992|loc=Theorem 9.6.}} $1.\; \backslash Rightarrow\; 3.$: Take p to be the natural surjection $V\; \backslash to\; V/W$. Since V is semisimple, p splits and so, through a section, $V/W$ is isomorphic to a subrepresentation that is complementary to W.

$3.\; \backslash Rightarrow\; 2.$: 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., $V\; =\; U\; \backslash oplus\; U\text{'}$ for some $U\text{'}$. By modular law, it implies $W\; =\; U\; \backslash oplus\; (W\; \backslash cap\; U\text{'})$. Then $(W\; \backslash cap\; U\text{'})\; \backslash simeq\; W/U$ is a simple subrepresentation

of W ("simple" because of maximality). This establishes the observation. Now, take $W$ to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation $W\text{'}$. If $W\text{'}\; \backslash ne\; 0$, then, by the early observation, $W\text{'}$ contains a simple subrepresentation and so $W\; \backslash cap\; W\text{'}\; \backslash ne\; 0$, a nonsense. Hence, $W\text{'}\; =\; 0$.

$2.\; \backslash Rightarrow\; 1.$:{{harvnb|Anderson|Fuller|1992|loc=Lemma 9.2.}} 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 $V\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; V\_i$ is a sum of simple subrepresentations, a semisimple decomposition $V\; =\; \backslash bigoplus\_\{i\backslash in\; I\text{'}\}\; V\_i$, some subset $I\text{'}\; \backslash subset\; I$, can be extracted from the sum.

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

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

For example, given a continuous finite-dimensional complex representation $\backslash pi:\; G\; \backslash to\; GL(V)$ of a finite group or a compact group G, by the averaging argument, one can define an inner product $\backslash langle\backslash ,\; ,\; \backslash rangle$ on V that is G-invariant: i.e., $\backslash langle\; \backslash pi(g)\; v,\; \backslash pi(g)\; w\; \backslash rangle\; =\; \backslash langle\; v,\; w\; \backslash rangle$, which is to say $\backslash pi(g)$ is a unitary operator and so $\backslash pi$ is a unitary representation.{{harvnb|Fulton|Harris|1991|loc=§ 9.3. A}} Hence, every finite-dimensional continuous complex representation of G is semisimple.{{harvnb|Hall|2015|loc=Theorem 4.28}} 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.{{sfn|Fulton|Harris|1991|loc=Corollary 1.6}}{{sfn|Serre|1977|loc=Theorem 2}}

By Weyl's theorem on complete reducibility, every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is semisimple.{{harvnb|Hall|2015}} Theorem 10.9

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.{{harvnb|Jacobson|1989|loc=§ 3.5. Exercise 4.}}

Given a finite-dimensional representation V, the Jordan–Hölder theorem says there is a filtration by subrepresentations: $V\; =\; V\_0\; \backslash supset\; V\_1\; \backslash supset\; \backslash cdots\; \backslash supset\; V\_n\; =\; 0$ such that each successive quotient $V\_i/V\_\{i+1\}$ is a simple representation. Then the associated vector space $\backslash operatorname\{gr\}\; V\; :=\; \backslash bigoplus\_\{i\; =\; 0\}^\{n\; -\; 1\}V\_i/V\_\{i+1\}$ is a semisimple representation called an associated semisimple representation, which, up to an isomorphism, is uniquely determined by V.{{harvnb|Artin|1999|loc=Ch. V, § 14.}}

A representation of a unipotent group is generally not semisimple. Take $G$ to be the group consisting of real matrices $\backslash begin\{bmatrix\}$

1 & a \\

0 & 1

\end{bmatrix}; it acts on $V\; =\; \backslash mathbb\{R\}^2$ 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 $\backslash begin\{bmatrix\}$

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).{{harvnb|Fulton|Harris|1991|loc=just after Corollary 1.6.}}

{{see also|Decomposition of a module}}

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.{{harvnb|Serre|1977|loc=§ 1.4. remark}} 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 isomorphism appearing in the decomposition of V (1) are unique and (2) completely determine the representation up to isomorphism;{{harvnb|Fulton|Harris|1991|loc=Proposition 1.8.}} 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):

:$V\; \backslash simeq\; \backslash bigoplus\_i\; V\_i^\{\backslash oplus\; m\_i\}$

where $V\_i$ are simple representations, mutually non-isomorphic to one another, and $m\_i$ are positive integers. By Schur's lemma,

:$m\_i\; =\; \backslash dim\; \backslash operatorname\{Hom\}\_\{\backslash text\{equiv\}\}(V\_i,\; V)\; =\; \backslash dim\; \backslash operatorname\{Hom\}\_\{\backslash text\{equiv\}\}(V,\; V\_i)$,

where $\backslash operatorname\{Hom\}\_\{\backslash text\{equiv\}\}$ 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 isomorphism, in V.{{harvnb|Fulton|Harris|1991|loc=§ 2.3.}}

In general, given a finite-dimensional representation $\backslash pi:\; G\; \backslash to\; GL(V)$ of a group G over a field k, the composition $\backslash chi\_V\; :\; G\; \backslash ,\backslash overset\{\backslash pi\}\{\backslash to\}\backslash ,\; GL(V)\; \backslash ,\backslash overset\{\backslash operatorname\{tr\}\}\{\backslash to\}\backslash ,\; k$ is called the character of $(\backslash pi,\; V)$.{{harvnb|Fulton|Harris|1991|loc=§ 2.1. Definition}} When $(\backslash pi,\; V)$ is semisimple with the decomposition $V\; \backslash simeq\; \backslash bigoplus\_i\; V\_i^\{\backslash oplus\; m\_i\}$ as above, the trace $\backslash operatorname\{tr\}(\backslash pi(g))$ is the sum of the traces of $\backslash pi(g)\; :\; V\_i\; \backslash to\; V\_i$ with multiplicities and thus, as functions on G,

:$\backslash chi\_V\; =\; \backslash sum\_i\; m\_i\; \backslash chi\_\{V\_i\}$

where $\backslash chi\_\{V\_i\}$ 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:{{harvnb|Serre|1977|loc=§ 2.3. Theorem 3 and § 4.3.}} the irreducible characters (characters of simple representations) of G are an orthonormal subset of the space of complex-valued functions on G and thus $m\_i\; =\; \backslash langle\; \backslash chi\_V,\; \backslash chi\_\{V\_i\}\; \backslash rangle$.

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;{{harvnb|Procesi|2007|loc=Ch. 6, § 2.3.}} 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:{{harvnb|Serre|1977|loc=§ 2.6. Theorem 8 (i)}}

:$V\; =\; \backslash bigoplus\_\{\backslash lambda\; \backslash in\; \backslash widehat\{G\}\}\; V^\{\backslash lambda\}$

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

The isotypic component of weight $\backslash lambda$ of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight $\backslash lambda$.

• A finite-dimensional module $V$ of a reductive Lie algebra $\backslash mathfrak\{g\}$ (or of the corresponding Lie group) can be decomposed into irreducible submodules

:: $V\; =\; \backslash bigoplus\_\{i=1\}^N\; V\_i$.

• Each finite-dimensional irreducible representation of $\backslash mathfrak\{g\}$ is uniquely identified (up to isomorphism) by its highest weight

:: $\backslash forall\; i\; \backslash in\; \backslash \{1,\backslash ldots,N\backslash \}\; \backslash ,\backslash exists\; \backslash lambda\; \backslash in\; P(\backslash mathfrak\{g\})\; :\; V\_i\; \backslash simeq\; M\_\backslash lambda$, where $M\_\backslash lambda$ denotes the highest weight module with highest weight $\backslash lambda$.

• In the decomposition of $V$, a certain isomorphism class might appear more than once, hence

:: $V\; \backslash simeq\; \backslash bigoplus\_\{\backslash lambda\; \backslash in\; P(\backslash mathfrak\{g\})\}\; (\backslash bigoplus\_\{i=1\}^\{d\_\backslash lambda\}\; M\_\{\backslash lambda\})$.

This defines the isotypic component of weight $\backslash lambda$ of $V$: $\backslash lambda(V)\; :=\; \backslash bigoplus\_\{i=1\}^\{d\_\backslash lambda\}\; V\_i\; \backslash simeq\; \backslash mathbb\{C\}^\{d\_\backslash lambda\}\; \backslash otimes\; M\_\{\backslash lambda\}$ where $d\_\backslash lambda$ is maximal.

Let $V$ be the space of homogeneous degree-three polynomials over the complex numbers in variables $x\_1,x\_2,x\_3$. Then $S\_3$ acts on $V$ 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 $S\_3$. In particular, $V$ contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation $W$ of $S\_3$. For example, the span of $x\_1^2x\_2-x\_2^2x\_1\; +\; x\_1^2x\_3-x\_2^2x\_3$ and $x\_2^2x\_3-x\_3^2x\_2\; +\; x\_2^2x\_1-x\_3^2x\_1$ is isomorphic to $W$. This can more easily be seen by writing this two-dimensional subspace as

:$W\_1=\backslash \{a(x\_1^2x\_2+x\_1^2x\_3)+b(x\_2^2x\_1+x\_2^2x\_3)+c(x\_3^2x\_1+x\_3^2x\_2)\backslash mid\; a+b+c\; =\; 0\backslash \}$.

Another copy of $W$ can be written in a similar form:

:$W\_2=\backslash \{a(x\_2^2x\_1+x\_3^2x\_1)+b(x\_1^2x\_2+x\_3^2x\_2)+c(x\_1^2x\_3+x\_2^2x\_3)\backslash mid\; a+b+c\; =\; 0\backslash \}$.

So can the third:

:$W\_3=\backslash \{ax\_1^3+bx\_2^3+cx\_3^3\backslash mid\; a+b+c\; =\; 0\backslash \}$.

Then $W\_1\; \backslash oplus\; W\_2\; \backslash oplus\; W\_3$ is the isotypic component of type $W$ in $V$.

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,{{harvnb|Procesi|2007|loc=Ch. 8, Theorem 3.2.}} there is a natural decomposition for $W\; =\; L^2(G)$ = the Hilbert space of (classes of) square-integrable functions on a compact group G:

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

where $\backslash widehat\{\backslash bigoplus\}$ means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations $(\backslash pi,\; V)$ of G.{{NoteTag|To be precise, the theorem concerns the regular representation of $G\; \backslash times\; G$ and the above statement is a corollary.}} 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 $W\; =\; \backslash mathbb\{C\}[G]$ is simply the group algebra of G and also the completion is vacuous. Thus, the theorem simply says that

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

That is, each simple representation of G appears in the regular representation with multiplicity the dimension of the representation.{{harvnb|Serre|1977|loc=§ 2.4. Corollary 1 to Proposition 5}} 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.{{harvnb|Procesi|2007|loc=Ch. 8, § 3.3.}}

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. 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.{{cite book |title = Quantum Theory for Mathematicians |chapter = Angular Momentum and Spin |pages=367–392 |last=Hall |first=Brian C. |publisher=Springer |series=Graduate Texts in Mathematics |volume=267 |year=2013 |isbn=978-1461471158 }} 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.{{cite journal |title = Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups |doi=10.1063/1.524268 |first1=A. U. |last1=Klimyk |first2=A. M. |last2=Gavrilik |year=1979 |journal=Journal of Mathematical Physics |volume=20 |issue=1624 |pages = 1624–1642 |bibcode=1979JMP....20.1624K }}

{{NoteFoot}}

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| publisher = Springer-Verlag

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Category:Representation theory