Semisimple representation

<h2 id="equivalent-characterizations">Equivalent characterizations</h2>
Let V be a representation of a group G; or more generally, let V be a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> with a set of linear <a href="/facts/Endomorphism/s7NNlSea">endomorphisms</a> acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be <a href="/facts/Simple_object/BgFLytBz">simple</a> (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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
The following are equivalent:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ol><li>V is semisimple as a representation.</li>
<li>V is a sum of simple <a href="/facts/Subrepresentation/MEJ9CMMD">subrepresentations</a>.</li>
<li>Each subrepresentation W of V admits a <a href="/facts/Complementary_representation/8uBU2Wqp">complementary representation</a>: a subrepresentation W' such that 
 
 
 
 V
 =
 W
 ⊕
 
 W
 ′
 
 
 
 {\displaystyle V=W\oplus W'}
 
.</li></ol>
The equivalence of the above conditions can be <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> based on the following <a href="/facts/Lemma_(mathematics)/zchvB1Ax">lemma</a>, which is of independent interest:

Lemma<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>—Let p:V → W be a <a href="/facts/Surjective/KezhLpCb">surjective</a> <a href="/facts/Representation_theory/3ydLtaGH">equivariant map</a> between representations. If V is semisimple, then p <a href="/facts/Splitting_lemma/L81DDNaA">splits</a>; i.e., it admits a <a href="/facts/Section_(category_theory)/Ee1wexPw">section</a>.

Proof of the lemma: Write 
 
 
 
 V
 =
 
 ⨁
 
 i
 ∈
 I
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V=\bigoplus _{i\in I}V_{i}}
 
 where 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
 are simple representations. <a href="/facts/Without_loss_of_generality/gVP4dmEe">Without loss of generality</a>, we can assume 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle 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
 
 
 :=
 
 ⨁
 
 i
 ∈
 J
 
 
 
 V
 
 i
 
 
 ⊂
 V
 
 
 {\displaystyle V_{J}:=\bigoplus _{i\in J}V_{i}\subset V}
 
 with various subsets 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
. Put the <a href="/facts/Partial_order/qb4a3FAX">partial ordering</a> on it by saying the direct sum over K is less than the direct sum over J if 
 
 
 
 K
 ⊂
 J
 
 
 {\displaystyle K\subset J}
 
. By <a href="/facts/Zorn%27s_lemma/ssCublFT">Zorn's lemma</a>, we can find a maximal 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
 such that 
 
 
 
 ker
 ⁡
 p
 ∩
 
 V
 
 J
 
 
 =
 0
 
 
 {\displaystyle \operatorname {ker} p\cap V_{J}=0}
 
. We claim that 
 
 
 
 V
 =
 ker
 ⁡
 p
 ⊕
 
 V
 
 J
 
 
 
 
 {\displaystyle V=\operatorname {ker} p\oplus V_{J}}
 
. By definition, 
 
 
 
 ker
 ⁡
 p
 ∩
 
 V
 
 J
 
 
 =
 0
 
 
 {\displaystyle \operatorname {ker} p\cap V_{J}=0}
 
 so we only need to show that 
 
 
 
 V
 =
 ker
 ⁡
 p
 +
 
 V
 
 J
 
 
 
 
 {\displaystyle V=\operatorname {ker} p+V_{J}}
 
. If 
 
 
 
 ker
 ⁡
 p
 +
 
 V
 
 J
 
 
 
 
 {\displaystyle \operatorname {ker} p+V_{J}}
 
 is a proper subrepresentatiom of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 then there exists 
 
 
 
 k
 ∈
 I
 −
 J
 
 
 {\displaystyle k\in I-J}
 
 such that 
 
 
 
 
 V
 
 k
 
 
 ⊄
 ker
 ⁡
 p
 +
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{k}\not \subset \operatorname {ker} p+V_{J}}
 
. Since 
 
 
 
 
 V
 
 k
 
 
 
 
 {\displaystyle V_{k}}
 
 is simple (irreducible), 
 
 
 
 
 V
 
 k
 
 
 ∩
 (
 ker
 ⁡
 p
 +
 
 V
 
 J
 
 
 )
 =
 0
 
 
 {\displaystyle V_{k}\cap (\operatorname {ker} p+V_{J})=0}
 
. This contradicts the maximality of 
 
 
 
 J
 
 
 {\displaystyle J}
 
, so 
 
 
 
 V
 =
 ker
 ⁡
 p
 ⊕
 
 V
 
 J
 
 
 
 
 {\displaystyle V=\operatorname {ker} p\oplus V_{J}}
 
 as claimed. Hence, 
 
 
 
 W
 ≃
 V
 
 /
 
 ker
 ⁡
 p
 ≃
 
 V
 
 J
 
 
 →
 V
 
 
 {\displaystyle W\simeq V/\operatorname {ker} p\simeq V_{J}\to V}
 
 is a section of p. 
 
 
 
 ◻
 
 
 {\displaystyle \square }

Note that we cannot take 
 
 
 
 J
 
 
 {\displaystyle J}
 
 to the set of 
 
 
 
 i
 
 
 {\displaystyle i}
 
 such that 
 
 
 
 ker
 ⁡
 (
 p
 )
 ∩
 
 V
 
 i
 
 
 =
 0
 
 
 {\displaystyle \ker(p)\cap V_{i}=0}
 
. The reason is that it can happen, and frequently does, that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a subspace of 
 
 
 
 Y
 ⊕
 Z
 
 
 {\displaystyle Y\oplus Z}
 
 and yet 
 
 
 
 X
 ∩
 Y
 =
 0
 =
 X
 ∩
 Z
 
 
 {\displaystyle X\cap Y=0=X\cap Z}
 
. For example, take 
 
 
 
 X
 
 
 {\displaystyle X}
 
, 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 to be three distinct lines through the origin in 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
. For an explicit counterexample, let 
 
 
 
 A
 =
 
 Mat
 
 2
 
 
 ⁡
 F
 
 
 {\displaystyle A=\operatorname {Mat} _{2}F}
 
 be the algebra of 2-by-2 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> and set 
 
 
 
 V
 =
 A
 
 
 {\displaystyle V=A}
 
, the regular representation of 
 
 
 
 A
 
 
 {\displaystyle A}
 
. Set 
 
 
 
 
 V
 
 1
 
 
 =
 
 
 {
 
 
 
 
 (
 
 
 
 a
 
 
 0
 
 
 
 
 b
 
 
 0
 
 
 
 )
 
 
 
 
 }
 
 
 
 
 {\displaystyle V_{1}={\Bigl \{}{\begin{pmatrix}a&0\\b&0\end{pmatrix}}{\Bigr \}}}
 
 and 
 
 
 
 
 V
 
 2
 
 
 =
 
 
 {
 
 
 
 
 (
 
 
 
 0
 
 
 c
 
 
 
 
 0
 
 
 d
 
 
 
 )
 
 
 
 
 }
 
 
 
 
 {\displaystyle V_{2}={\Bigl \{}{\begin{pmatrix}0&c\\0&d\end{pmatrix}}{\Bigr \}}}
 
 and set 
 
 
 
 W
 =
 
 
 {
 
 
 
 
 (
 
 
 
 c
 
 
 c
 
 
 
 
 d
 
 
 d
 
 
 
 )
 
 
 
 
 }
 
 
 
 
 {\displaystyle W={\Bigl \{}{\begin{pmatrix}c&c\\d&d\end{pmatrix}}{\Bigr \}}}
 
. Then 
 
 
 
 
 V
 
 1
 
 
 
 
 {\displaystyle V_{1}}
 
, 
 
 
 
 
 V
 
 2
 
 
 
 
 {\displaystyle V_{2}}
 
 and 
 
 
 
 W
 
 
 {\displaystyle W}
 
 are all <a href="/facts/Irreducible_module/00roKQrG">irreducible 
 
 
 
 A
 
 
 {\displaystyle A}
 
-modules</a> and 
 
 
 
 V
 =
 
 V
 
 1
 
 
 ⊕
 
 V
 
 2
 
 
 
 
 {\displaystyle V=V_{1}\oplus V_{2}}
 
. Let 
 
 
 
 p
 :
 V
 →
 V
 
 /
 
 W
 
 
 {\displaystyle p:V\to V/W}
 
 be the natural surjection. Then 
 
 
 
 ker
 ⁡
 p
 =
 W
 ≠
 0
 
 
 {\displaystyle \operatorname {ker} p=W\neq 0}
 
 and 
 
 
 
 
 V
 
 1
 
 
 ∩
 ker
 ⁡
 p
 =
 0
 =
 
 V
 
 2
 
 
 ∩
 ker
 ⁡
 p
 
 
 {\displaystyle V_{1}\cap \operatorname {ker} p=0=V_{2}\cap \operatorname {ker} p}
 
. In this case, 
 
 
 
 W
 ≃
 
 V
 
 1
 
 
 ≃
 
 V
 
 2
 
 
 
 
 {\displaystyle W\simeq V_{1}\simeq V_{2}}
 
 but 
 
 
 
 V
 ≠
 ker
 ⁡
 p
 ⊕
 
 V
 
 1
 
 
 ⊕
 
 V
 
 2
 
 
 
 
 {\displaystyle V\neq \operatorname {ker} p\oplus V_{1}\oplus V_{2}}
 
 because this sum is not direct.
Proof of equivalences<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 
 
 
 
 1.
 ⇒
 3.
 
 
 {\displaystyle 1.\Rightarrow 3.}
 
: Take p to be the natural surjection 
 
 
 
 V
 →
 V
 
 /
 
 W
 
 
 {\displaystyle V\to V/W}
 
. Since V is semisimple, p splits and so, through a section, 
 
 
 
 V
 
 /
 
 W
 
 
 {\displaystyle V/W}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to a subrepresentation that is complementary to W.

 
 
 
 3.
 ⇒
 2.
 
 
 {\displaystyle 3.\Rightarrow 2.}
 
: We shall first observe that every nonzero subrepresentation W has a simple subrepresentation. Shrinking W to a (nonzero) <a href="/facts/Cyclic_module/w9tMplLC">cyclic subrepresentation</a> we can assume it is finitely generated. Then it has a <a href="/facts/Maximal_submodule/nf4DnKVS">maximal subrepresentation</a> U. By the condition 3., 
 
 
 
 V
 =
 U
 ⊕
 
 U
 ′
 
 
 
 {\displaystyle V=U\oplus U'}
 
 for some 
 
 
 
 
 U
 ′
 
 
 
 {\displaystyle U'}
 
. By modular law, it implies 
 
 
 
 W
 =
 U
 ⊕
 (
 W
 ∩
 
 U
 ′
 
 )
 
 
 {\displaystyle W=U\oplus (W\cap U')}
 
. Then 
 
 
 
 (
 W
 ∩
 
 U
 ′
 
 )
 ≃
 W
 
 /
 
 U
 
 
 {\displaystyle (W\cap U')\simeq W/U}
 
 is a simple subrepresentation
of W ("simple" because of maximality). This establishes the observation. Now, take 
 
 
 
 W
 
 
 {\displaystyle W}
 
 to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation 
 
 
 
 
 W
 ′
 
 
 
 {\displaystyle W'}
 
. If 
 
 
 
 
 W
 ′
 
 ≠
 0
 
 
 {\displaystyle W'\neq 0}
 
, then, by the early observation, 
 
 
 
 
 W
 ′
 
 
 
 {\displaystyle W'}
 
 contains a simple subrepresentation and so 
 
 
 
 W
 ∩
 
 W
 ′
 
 ≠
 0
 
 
 {\displaystyle W\cap W'\neq 0}
 
, a nonsense. Hence, 
 
 
 
 
 W
 ′
 
 =
 0
 
 
 {\displaystyle W'=0}
 
.

 
 
 
 2.
 ⇒
 1.
 
 
 {\displaystyle 2.\Rightarrow 1.}
 
:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 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:

<ul><li>When 
 
 
 
 V
 =
 
 ∑
 
 i
 ∈
 I
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V=\sum _{i\in I}V_{i}}
 
 is a sum of simple subrepresentations, a semisimple decomposition 
 
 
 
 V
 =
 
 ⨁
 
 i
 ∈
 
 I
 ′
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V=\bigoplus _{i\in I'}V_{i}}
 
, some subset 
 
 
 
 
 I
 ′
 
 ⊂
 I
 
 
 {\displaystyle I'\subset I}
 
, can be extracted from the sum.</li></ul>
As in the proof of the lemma, we can find a maximal direct sum 
 
 
 
 W
 
 
 {\displaystyle W}
 
 that consists of some 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
's. Now, for each i in I, by simplicity, either 
 
 
 
 
 V
 
 i
 
 
 ⊂
 W
 
 
 {\displaystyle V_{i}\subset W}
 
 or 
 
 
 
 
 V
 
 i
 
 
 ∩
 W
 =
 0
 
 
 {\displaystyle V_{i}\cap W=0}
 
. In the second case, the direct sum 
 
 
 
 W
 ⊕
 
 V
 
 i
 
 
 
 
 {\displaystyle W\oplus V_{i}}
 
 is a contradiction to the maximality of W. Hence, 
 
 
 
 
 V
 
 i
 
 
 ⊂
 W
 
 
 {\displaystyle V_{i}\subset W}
 
. 
 
 
 
 ◻
 
 
 {\displaystyle \square }

<h2 id="examples-and-non-examples">Examples and non-examples</h2>
<h3>Unitary representations</h3>
A finite-dimensional <a href="/facts/Unitary_representation/W1XRatG8">unitary representation</a> (i.e., a representation factoring through a <a href="/facts/Unitary_group/lEvudIhU">unitary group</a>) 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<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> because if 
 
 
 
 v
 ∈
 
 W
 
 ⊥
 
 
 
 
 {\displaystyle v\in W^{\bot }}
 
 and 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
, then 
 
 
 
 ⟨
 π
 (
 g
 )
 v
 ,
 w
 ⟩
 =
 ⟨
 v
 ,
 π
 (
 
 g
 
 −
 1
 
 
 )
 w
 ⟩
 =
 0
 
 
 {\displaystyle \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 
 
 
 
 π
 (
 g
 )
 v
 ∈
 
 W
 
 ⊥
 
 
 
 
 {\displaystyle \pi (g)v\in W^{\bot }}
 
.
For example, given a continuous finite-dimensional <a href="/facts/Complex_number/2w7mqI7e">complex</a> representation 
 
 
 
 π
 :
 G
 →
 G
 L
 (
 V
 )
 
 
 {\displaystyle \pi :G\to GL(V)}
 
 of a <a href="/facts/Finite_group/vv2tvogB">finite group</a> or a <a href="/facts/Compact_group/Itl2HJaR">compact group</a> G, by the averaging argument, one can define an <a href="/facts/Inner_product/HoyjElEy">inner product</a> 
 
 
 
 ⟨
 
 ,
 ⟩
 
 
 {\displaystyle \langle \,,\rangle }
 
 on V that is G-invariant: i.e., 
 
 
 
 ⟨
 π
 (
 g
 )
 v
 ,
 π
 (
 g
 )
 w
 ⟩
 =
 ⟨
 v
 ,
 w
 ⟩
 
 
 {\displaystyle \langle \pi (g)v,\pi (g)w\rangle =\langle v,w\rangle }
 
, which is to say 
 
 
 
 π
 (
 g
 )
 
 
 {\displaystyle \pi (g)}
 
 is a unitary operator and so 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 is a unitary representation.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Hence, every finite-dimensional continuous complex representation of G is semisimple.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> For a finite group, this is a special case of <a href="/facts/Maschke%27s_theorem/uFMFeyfA">Maschke's theorem</a>, which says a finite-dimensional representation of a finite group G over a field k with <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> not dividing the <a href="/facts/Order_of_a_group/Cl3tKGFg">order</a> of G is semisimple.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h3>Representations of semisimple Lie algebras</h3>
By <a href="/facts/Weyl%27s_theorem_on_complete_reducibility/pNRfa6Nu">Weyl's theorem on complete reducibility</a>, every finite-dimensional representation of a <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">semisimple Lie algebra</a> over a field of characteristic zero is semisimple.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h3>Separable minimal polynomials</h3>
Given a linear endomorphism T of a vector space V, V is semisimple as a representation of T (i.e., T is a <a href="/facts/Semisimple_operator/J3Jdl8Qh">semisimple operator</a>) <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<h3>Associated semisimple representation</h3>
Given a finite-dimensional representation V, the <a href="/facts/Jordan%E2%80%93H%C3%B6lder_theorem/3C7CHH3k">Jordan–Hölder theorem</a> says there is a filtration by subrepresentations: 
 
 
 
 V
 =
 
 V
 
 0
 
 
 ⊃
 
 V
 
 1
 
 
 ⊃
 ⋯
 ⊃
 
 V
 
 n
 
 
 =
 0
 
 
 {\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0}
 
 such that each successive quotient 
 
 
 
 
 V
 
 i
 
 
 
 /
 
 
 V
 
 i
 +
 1
 
 
 
 
 {\displaystyle V_{i}/V_{i+1}}
 
 is a simple representation. Then the associated vector space 
 
 
 
 gr
 ⁡
 V
 :=
 
 ⨁
 
 i
 =
 0
 
 
 n
 −
 1
 
 
 
 V
 
 i
 
 
 
 /
 
 
 V
 
 i
 +
 1
 
 
 
 
 {\displaystyle \operatorname {gr} V:=\bigoplus _{i=0}^{n-1}V_{i}/V_{i+1}}
 
 is a semisimple representation called an associated semisimple representation, which, <a href="/facts/Up_to/ydz4ztzX">up to</a> an isomorphism, is uniquely determined by V.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h3>Unipotent group non-example</h3>
A representation of a <a href="/facts/Unipotent_group/xnJlzBR9">unipotent group</a> is generally not semisimple. Take 
 
 
 
 G
 
 
 {\displaystyle G}
 
 to be the group consisting of <a href="/facts/Real_number/R02gw5Pb">real</a> matrices 
 
 
 
 
 
 [
 
 
 
 1
 
 
 a
 
 
 
 
 0
 
 
 1
 
 
 
 ]
 
 
 
 
 {\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}}
 
; it acts on 
 
 
 
 V
 =
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle V=\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 
 
 
 
 
 
 [
 
 
 
 1
 
 
 
 
 0
 
 
 
 ]
 
 
 
 
 {\displaystyle {\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).<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h2 id="semisimple-decomposition-and-multiplicity">Semisimple decomposition and multiplicity</h2>
See also: <a href="/facts/Decomposition_of_a_module/9ujxGNUW">Decomposition of a module</a>
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.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> The isotypic decomposition, on the other hand, is an example of a unique decomposition.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
However, for a finite-dimensional semisimple representation V over an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a>, 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;<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> this is a consequence of <a href="/facts/Schur%27s_lemma/JvAT7t09">Schur's lemma</a> 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):<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

V
        ≃
        
          ⨁
          
            i
          
        
        
          V
          
            i
          
          
            ⊕
            
              m
              
                i
              
            
          
        
      
    
    {\displaystyle V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}}

where 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
 are simple representations, mutually non-isomorphic to one another, and 
 
 
 
 
 m
 
 i
 
 
 
 
 {\displaystyle m_{i}}
 
 are positive <a href="/facts/Integer/PUwwrawx">integers</a>. By Schur's lemma,

m
 
 i
 
 
 =
 dim
 ⁡
 
 Hom
 
 equiv
 
 
 ⁡
 (
 
 V
 
 i
 
 
 ,
 V
 )
 =
 dim
 ⁡
 
 Hom
 
 equiv
 
 
 ⁡
 (
 V
 ,
 
 V
 
 i
 
 
 )
 
 
 {\displaystyle m_{i}=\dim \operatorname {Hom} _{\text{equiv}}(V_{i},V)=\dim \operatorname {Hom} _{\text{equiv}}(V,V_{i})}
 
,
where 
 
 
 
 
 Hom
 
 equiv
 
 
 
 
 {\displaystyle \operatorname {Hom} _{\text{equiv}}}
 
 refers to the <a href="/facts/Equivariant_map/KkIkDwIH">equivariant linear maps</a>. Also, each 
 
 
 
 
 m
 
 i
 
 
 
 
 {\displaystyle m_{i}}
 
 is unchanged if 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
 is replaced by another simple representation isomorphic to 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
. Thus, the integers 
 
 
 
 
 m
 
 i
 
 
 
 
 {\displaystyle m_{i}}
 
 are independent of chosen decompositions; they are the multiplicities of simple representations 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
, up to isomorphism, in V.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
In general, given a finite-dimensional representation 
 
 
 
 π
 :
 G
 →
 G
 L
 (
 V
 )
 
 
 {\displaystyle \pi :G\to GL(V)}
 
 of a group G over a field k, the composition 
 
 
 
 
 χ
 
 V
 
 
 :
 G
 
 
 
 →
 π
 
 
 
 G
 L
 (
 V
 )
 
 
 
 →
 tr
 
 
 
 k
 
 
 {\displaystyle \chi _{V}:G\,{\overset {\pi }{\to }}\,GL(V)\,{\overset {\operatorname {tr} }{\to }}\,k}
 
 is called the <a href="/facts/Character_of_a_representation/WMp5Q6YE">character</a> of 
 
 
 
 (
 π
 ,
 V
 )
 
 
 {\displaystyle (\pi ,V)}
 
.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> When 
 
 
 
 (
 π
 ,
 V
 )
 
 
 {\displaystyle (\pi ,V)}
 
 is semisimple with the decomposition 
 
 
 
 V
 ≃
 
 ⨁
 
 i
 
 
 
 V
 
 i
 
 
 ⊕
 
 m
 
 i
 
 
 
 
 
 
 {\displaystyle V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}}
 
 as above, the trace 
 
 
 
 tr
 ⁡
 (
 π
 (
 g
 )
 )
 
 
 {\displaystyle \operatorname {tr} (\pi (g))}
 
 is the sum of the traces of 
 
 
 
 π
 (
 g
 )
 :
 
 V
 
 i
 
 
 →
 
 V
 
 i
 
 
 
 
 {\displaystyle \pi (g):V_{i}\to V_{i}}
 
 with multiplicities and thus, as functions on G,

χ
          
            V
          
        
        =
        
          ∑
          
            i
          
        
        
          m
          
            i
          
        
        
          χ
          
            
              V
              
                i
              
            
          
        
      
    
    {\displaystyle \chi _{V}=\sum _{i}m_{i}\chi _{V_{i}}}

where 
 
 
 
 
 χ
 
 
 V
 
 i
 
 
 
 
 
 
 {\displaystyle \chi _{V_{i}}}
 
 are the characters of 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V_{i}}
 
. When G is a finite group or more generally a compact group and 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a unitary representation with the inner product given by the averaging argument, the <a href="/facts/Schur_orthogonality_relations/K70B54z9">Schur orthogonality relations</a> say:<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> 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
 
 
 =
 ⟨
 
 χ
 
 V
 
 
 ,
 
 χ
 
 
 V
 
 i
 
 
 
 
 ⟩
 
 
 {\displaystyle m_{i}=\langle \chi _{V},\chi _{V_{i}}\rangle }
 
.

<h2 id="isotypic-decomposition">Isotypic decomposition</h2>
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 <a href="/facts/Isotypic_component/HR1fDx7k">isotypic component</a> of type S of a representation V is the sum of all subrepresentations of V that are isomorphic to S;<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> 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:<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

V
        =
        
          ⨁
          
            λ
            ∈
            
              
                
                  G
                  ^
                
              
            
          
        
        
          V
          
            λ
          
        
      
    
    {\displaystyle V=\bigoplus _{\lambda \in {\widehat {G}}}V^{\lambda }}

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

<h3>Isotypic component</h3>
The isotypic component of <a href="/facts/Weight_(representation_theory)/7N8q5sw0">weight</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of a <a href="/facts/Lie_algebra_module/aaxnMxCR">Lie algebra module</a> is the sum of all submodules which are <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the highest weight module with weight 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
.

<h4>Definition</h4>
<ul><li>A finite-dimensional <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of a <a href="/facts/Reductive_Lie_algebra/RpYjWVvM">reductive Lie algebra</a> 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 (or of the corresponding <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>) can be decomposed into <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible submodules</a></li></ul>

V
 =
 
 ⨁
 
 i
 =
 1
 
 
 N
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V=\bigoplus _{i=1}^{N}V_{i}}
 
.
<ul><li>Each finite-dimensional irreducible representation of 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 is uniquely identified (up to isomorphism) by its <a href="/facts/Weight_(representation_theory)/7N8q5sw0">highest weight</a></li></ul>

∀
 i
 ∈
 {
 1
 ,
 …
 ,
 N
 }
 
 ∃
 λ
 ∈
 P
 (
 
 
 g
 
 
 )
 :
 
 V
 
 i
 
 
 ≃
 
 M
 
 λ
 
 
 
 
 {\displaystyle \forall i\in \{1,\ldots ,N\}\,\exists \lambda \in P({\mathfrak {g}}):V_{i}\simeq M_{\lambda }}
 
, where 
 
 
 
 
 M
 
 λ
 
 
 
 
 {\displaystyle M_{\lambda }}
 
 denotes the <a href="/facts/Weight_(representation_theory)/7N8q5sw0">highest weight module</a> with highest weight 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
.
<ul><li>In the decomposition of 
 
 
 
 V
 
 
 {\displaystyle V}
 
, a certain <a href="/facts/Isomorphism_class/wDUgsJQ8">isomorphism class</a> might appear more than once, hence</li></ul>

V
 ≃
 
 ⨁
 
 λ
 ∈
 P
 (
 
 
 g
 
 
 )
 
 
 (
 
 ⨁
 
 i
 =
 1
 
 
 
 d
 
 λ
 
 
 
 
 
 M
 
 λ
 
 
 )
 
 
 {\displaystyle V\simeq \bigoplus _{\lambda \in P({\mathfrak {g}})}(\bigoplus _{i=1}^{d_{\lambda }}M_{\lambda })}
 
.
This defines the isotypic component of weight 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 of 
 
 
 
 V
 
 
 {\displaystyle V}
 
: 
 
 
 
 λ
 (
 V
 )
 :=
 
 ⨁
 
 i
 =
 1
 
 
 
 d
 
 λ
 
 
 
 
 
 V
 
 i
 
 
 ≃
 
 
 C
 
 
 
 d
 
 λ
 
 
 
 
 ⊗
 
 M
 
 λ
 
 
 
 
 {\displaystyle \lambda (V):=\bigoplus _{i=1}^{d_{\lambda }}V_{i}\simeq \mathbb {C} ^{d_{\lambda }}\otimes M_{\lambda }}
 
 where 
 
 
 
 
 d
 
 λ
 
 
 
 
 {\displaystyle d_{\lambda }}
 
 is maximal.

<h3>Example</h3>
Let 
 
 
 
 V
 
 
 {\displaystyle V}
 
 be the space of homogeneous degree-three polynomials over the complex numbers in variables 
 
 
 
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 
 x
 
 3
 
 
 
 
 {\displaystyle x_{1},x_{2},x_{3}}
 
. Then 
 
 
 
 
 S
 
 3
 
 
 
 
 {\displaystyle S_{3}}
 
 acts on 
 
 
 
 V
 
 
 {\displaystyle 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
 
 
 
 
 {\displaystyle S_{3}}
 
. In particular, 
 
 
 
 V
 
 
 {\displaystyle V}
 
 contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation 
 
 
 
 W
 
 
 {\displaystyle W}
 
 of 
 
 
 
 
 S
 
 3
 
 
 
 
 {\displaystyle S_{3}}
 
. For example, the span of 
 
 
 
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 2
 
 
 −
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 1
 
 
 +
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 3
 
 
 −
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 3
 
 
 
 
 {\displaystyle x_{1}^{2}x_{2}-x_{2}^{2}x_{1}+x_{1}^{2}x_{3}-x_{2}^{2}x_{3}}
 
 and 
 
 
 
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 3
 
 
 −
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 2
 
 
 +
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 1
 
 
 −
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 1
 
 
 
 
 {\displaystyle x_{2}^{2}x_{3}-x_{3}^{2}x_{2}+x_{2}^{2}x_{1}-x_{3}^{2}x_{1}}
 
 is isomorphic to 
 
 
 
 W
 
 
 {\displaystyle W}
 
. This can more easily be seen by writing this two-dimensional subspace as

W
 
 1
 
 
 =
 {
 a
 (
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 2
 
 
 +
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 3
 
 
 )
 +
 b
 (
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 1
 
 
 +
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 3
 
 
 )
 +
 c
 (
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 1
 
 
 +
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 2
 
 
 )
 ∣
 a
 +
 b
 +
 c
 =
 0
 }
 
 
 {\displaystyle W_{1}=\{a(x_{1}^{2}x_{2}+x_{1}^{2}x_{3})+b(x_{2}^{2}x_{1}+x_{2}^{2}x_{3})+c(x_{3}^{2}x_{1}+x_{3}^{2}x_{2})\mid a+b+c=0\}}
 
.
Another copy of 
 
 
 
 W
 
 
 {\displaystyle W}
 
 can be written in a similar form:

W
 
 2
 
 
 =
 {
 a
 (
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 1
 
 
 +
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 1
 
 
 )
 +
 b
 (
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 2
 
 
 +
 
 x
 
 3
 
 
 2
 
 
 
 x
 
 2
 
 
 )
 +
 c
 (
 
 x
 
 1
 
 
 2
 
 
 
 x
 
 3
 
 
 +
 
 x
 
 2
 
 
 2
 
 
 
 x
 
 3
 
 
 )
 ∣
 a
 +
 b
 +
 c
 =
 0
 }
 
 
 {\displaystyle W_{2}=\{a(x_{2}^{2}x_{1}+x_{3}^{2}x_{1})+b(x_{1}^{2}x_{2}+x_{3}^{2}x_{2})+c(x_{1}^{2}x_{3}+x_{2}^{2}x_{3})\mid a+b+c=0\}}
 
.
So can the third:

W
 
 3
 
 
 =
 {
 a
 
 x
 
 1
 
 
 3
 
 
 +
 b
 
 x
 
 2
 
 
 3
 
 
 +
 c
 
 x
 
 3
 
 
 3
 
 
 ∣
 a
 +
 b
 +
 c
 =
 0
 }
 
 
 {\displaystyle W_{3}=\{ax_{1}^{3}+bx_{2}^{3}+cx_{3}^{3}\mid a+b+c=0\}}
 
.
Then 
 
 
 
 
 W
 
 1
 
 
 ⊕
 
 W
 
 2
 
 
 ⊕
 
 W
 
 3
 
 
 
 
 {\displaystyle W_{1}\oplus W_{2}\oplus W_{3}}
 
 is the isotypic component of type 
 
 
 
 W
 
 
 {\displaystyle W}
 
 in 
 
 
 
 V
 
 
 {\displaystyle V}
 
.

<h2 id="completion">Completion</h2>
In <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a>, 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 <a href="/facts/Peter%E2%80%93Weyl_theorem/vQfn6cQr">Peter–Weyl theorem</a>, which decomposes the left (or right) <a href="/facts/Regular_representation/dBND2lRj">regular representation</a> of a compact group into the Hilbert-space completion of the direct sum of all simple unitary representations. As a <a href="/facts/Corollary/ntQ8zR9N">corollary</a>,<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> there is a natural decomposition for 
 
 
 
 W
 =
 
 L
 
 2
 
 
 (
 G
 )
 
 
 {\displaystyle W=L^{2}(G)}
 
 = the Hilbert space of (classes of) square-integrable functions on a compact group G:

W
        ≃
        
          
            
              
                ⨁
                
                  [
                  (
                  π
                  ,
                  V
                  )
                  ]
                
              
              ^
            
          
        
        
          V
          
            ⊕
            dim
            ⁡
            V
          
        
      
    
    {\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 
 
 
 
 (
 π
 ,
 V
 )
 
 
 {\displaystyle (\pi ,V)}
 
 of G.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> 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
 =
 
 C
 
 [
 G
 ]
 
 
 {\displaystyle W=\mathbb {C} [G]}
 
 is simply the group algebra of G and also the completion is vacuous. Thus, the theorem simply says that

C
        
        [
        G
        ]
        =
        
          ⨁
          
            [
            (
            π
            ,
            V
            )
            ]
          
        
        
          V
          
            ⊕
            dim
            ⁡
            V
          
        
        .
      
    
    {\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.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> 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 <a href="/facts/Circle_group/kuzdWtHo">circle group</a> 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
, the theorem exactly amounts to the classical Fourier analysis.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>

<h2 id="applications-to-physics">Applications to physics</h2>
In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> and <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>, the <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> of an object can be described by <a href="/facts/Representation_of_a_Lie_group/29R96ar2">complex representations of the rotation group SO(3)</a>, all of which are semisimple.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> Due to <a href="/facts/3D_rotation_group/yXbKPzhT">connection between SO(3) and SU(2)</a>, the non-relativistic <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a> of an <a href="/facts/Elementary_particle/CTFDM97U">elementary particle</a> is described by <a href="/facts/Representation_theory_of_SU(2)/fBw5fr2r">complex representations of SU(2)</a> and the relativistic spin is described by <a href="/facts/Representation_theory_of_the_Lorentz_group/XJRuPVkL">complex representations of SL2(C)</a>, all of which are semisimple.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> In <a href="/facts/Angular_momentum_coupling/xsfw1hBG">angular momentum coupling</a>, <a href="/facts/Clebsch%E2%80%93Gordan_coefficients/r063mruI">Clebsch–Gordan coefficients</a> arise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>

<h2 id="notes">Notes</h2>

<h3>Citations</h3>

<h3>Sources</h3>

<ul><li>Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 13 (2nd ed.), New York, NY: Springer-Verlag, pp. x+376, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-4418-9">10.1007/978-1-4612-4418-9</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97845-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1245487">1245487</a>; 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).</li>
<li>Artin, Michael (1999). <a href="http://math.mit.edu/~etingof/artinnotes.pdf">"Noncommutative Rings"</a> (PDF).</li>
<li><a href="/facts/William_Fulton_(mathematician)/8ubLHaPy">Fulton, William</a>; <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Harris, Joe</a> (1991). Representation theory. A first course. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0979-9">10.1007/978-1-4612-0979-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97495-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153249">1153249</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/246650103">246650103</a>.</li>
<li>Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3319134666.</li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (1989), Basic algebra II (2nd ed.), W. H. Freeman, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7167-1933-5</li>
<li><a href="/facts/Claudio_Procesi/327Bhv6R">Procesi, Claudio</a> (2007). Lie Groups: an approach through invariants and representation. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780387260402..</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1977-09-01). <a href="https://archive.org/details/linearrepresenta1977serr">Linear Representations of Finite Groups</a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, 42. New York–Heidelberg: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90190-9. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0450380">0450380</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0355.20006">0355.20006</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Procesi 2007, Ch. 6, § 1.1, Definition 1 (ii). - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Procesi 2007, Ch. 6, § 1.1, Definition 1 (ii). - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Procesi 2007, Ch. 6, § 2.1. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Anderson & Fuller 1992, Proposition 9.4. - Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2nd ed.), New York, NY: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 <a href="https://doi.org/10.1007%2F978-1-4612-4418-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-4418-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Anderson & Fuller 1992, Theorem 9.6. - Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2nd ed.), New York, NY: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 <a href="https://doi.org/10.1007%2F978-1-4612-4418-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-4418-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Anderson & Fuller 1992, Lemma 9.2. - Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2nd ed.), New York, NY: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 <a href="https://doi.org/10.1007%2F978-1-4612-4418-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-4418-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Fulton & Harris 1991, § 9.3. A - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Fulton & Harris 1991, § 9.3. A - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hall 2015, Theorem 4.28 - Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. ISBN 978-3319134666. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Fulton & Harris 1991, Corollary 1.6. - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Serre 1977, Theorem 2. - 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. <a href="https://archive.org/details/linearrepresenta1977serr" target="_blank">https://archive.org/details/linearrepresenta1977serr</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Hall 2015 Theorem 10.9 - Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. ISBN 978-3319134666. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Jacobson 1989, § 3.5. Exercise 4. - Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5 <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Artin 1999, Ch. V, § 14. - Artin, Michael (1999). "Noncommutative Rings" (PDF). <a href="http://math.mit.edu/~etingof/artinnotes.pdf" target="_blank">http://math.mit.edu/~etingof/artinnotes.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Fulton & Harris 1991, just after Corollary 1.6. - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Serre 1977, § 1.4. remark - 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. <a href="https://archive.org/details/linearrepresenta1977serr" target="_blank">https://archive.org/details/linearrepresenta1977serr</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Procesi 2007, Ch. 6, § 2.3. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Fulton & Harris 1991, Proposition 1.8. - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Fulton & Harris 1991, Proposition 1.8. - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Fulton & Harris 1991, § 2.3. - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Fulton & Harris 1991, § 2.1. Definition - Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. <a href="https://doi.org/10.1007%2F978-1-4612-0979-9" target="_blank">https://doi.org/10.1007%2F978-1-4612-0979-9</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Serre 1977, § 2.3. Theorem 3 and § 4.3. - 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. <a href="https://archive.org/details/linearrepresenta1977serr" target="_blank">https://archive.org/details/linearrepresenta1977serr</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Procesi 2007, Ch. 6, § 2.3. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Procesi 2007, Ch. 6, § 2.3. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Serre 1977, § 2.6. Theorem 8 (i) - 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. <a href="https://archive.org/details/linearrepresenta1977serr" target="_blank">https://archive.org/details/linearrepresenta1977serr</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Procesi 2007, Ch. 8, Theorem 3.2. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">To be precise, the theorem concerns the regular representation of 
 
 
 
 G
 ×
 G
 
 
 {\displaystyle G\times G}
 
 and the above statement is a corollary. <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Serre 1977, § 2.4. Corollary 1 to Proposition 5 - 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. <a href="https://archive.org/details/linearrepresenta1977serr" target="_blank">https://archive.org/details/linearrepresenta1977serr</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Procesi 2007, Ch. 8, § 3.3. - Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Hall, Brian C. (2013). "Angular Momentum and Spin". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. Springer. pp. 367–392. ISBN 978-1461471158. <a href="978-1461471158" target="_blank">978-1461471158</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Hall, Brian C. (2013). "Angular Momentum and Spin". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. Springer. pp. 367–392. ISBN 978-1461471158. <a href="978-1461471158" target="_blank">978-1461471158</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Klimyk, A. U.; Gavrilik, A. M. (1979). "Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups". Journal of Mathematical Physics. 20 (1624): 1624–1642. Bibcode:1979JMP....20.1624K. doi:10.1063/1.524268. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
</ol>

Semisimple representation open-in-new

Semisimple representation