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).[1] 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.[1]

The following are equivalent:[2]

V is semisimple as a representation.
V is a sum of simple subrepresentations.
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[3] — 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 lemma —

Write $V=\bigoplus _{i}V_{i}$ 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. By simplicity, either $V_{i}\subset \operatorname {ker} (p)$ or $\operatorname {ker} (p)\cap V_{i}=0$ . Thus, $V=\operatorname {ker} (p)\bigoplus \left(\bigoplus _{i\in I'}V_{i}\right)$ where $I'$ is such that for each $i\in I'$ , $V_{i}\cap \operatorname {ker} (p)=0$ . Then $W\simeq \bigoplus _{i\in I'}V_{i}\to V$ is a section of p. $\square$

Proof of equivalences[4] —

$1.\Rightarrow 3.$ : Take p to be the natural surjection $V\to V/W$ . Since V is semisimple, p splits and so, through a section, $V/W$ is isomorphic to a subrepretation that is complementary to W.

$3.\Rightarrow 2.$ : 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., $V=U\oplus U'$ for some $U'$ . By modular law, it implies $W=U\oplus (W\cap U')$ . Then $(W\cap U')\simeq W/U$ 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 $W'$ . If $W'\neq 0$ , then, by the early observation, $W'$ contains a simple subrepresentation and so $W\cap W'\neq 0$ , a nonsense. Hence, $W'=0$ .

$2.\Rightarrow 1.$ :[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 the statement we can show is: when $V=\sum _{i\in I}V_{i}$ is a sum of simple subrepresentations, a semisimple decomposition $V=\bigoplus _{i\in I'}V_{i}$ , some subset $I'\subset I$ , can be extracted from the sum. Consider the family of all possible direct sums $\bigoplus _{i\in J}V_{i}\subset V$ with various subsets $J\subset I$ . Put the partial ordering on it by saying the direct sum over K is less than the direct sum over J if $K\subset J$ . Zorn's lemma clearly applies to it and gives us a maximal direct sum W. Now, for each i in I, by simplicity, either $V_{i}\subset W$ or $V_{i}\cap W=0$ . In the second case, the direct sum $W\oplus V_{i}$ is a contradiction to the maximality of W. Hence, $V_{i}\subset W$ . $\square$

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[6] 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.[6] Hence, every finite-dimensional continuous complex representation of G is semisimple.[7] 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.[8][9]

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.[10]

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.[11]

Associated semisimple representation

Given a finite-dimensional representation V, the Jordan–Hölder theorem says there is a filtration by subrepresentations: $V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0$ such that each successive quotient $V_{i}/V_{i+1}$ is a simple representation. Then the associated vector space $\operatorname {gr} V:=\bigoplus _{i=0}^{n-1}V_{i}/V_{i+1}$ is a semisimple representation called an associated semisimple representation, which, up to an isomorphism, is uniquely determined by V.[12]

Unipotent group non-example

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 on $V=\mathbb {R} ^{2}$ 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 ${\begin{bmatrix}1\\0\end{bmatrix}}$ . 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).[13]

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.[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]

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.[17]

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 $(\pi ,V)$ .[18] 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:[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 $m_{i}=\langle \chi _{V},\chi _{V_{i}}\rangle$ .

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;[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]

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 $S\in \lambda$ .

Example

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 $x_{1}^{2}x_{2}-x_{2}^{2}x_{1}+x_{1}^{2}x_{3}-x_{2}^{2}x_{3}$ and $x_{2}^{2}x_{3}-x_{3}^{2}x_{2}+x_{2}^{2}x_{1}-x_{3}^{2}x_{1}$ is isomorphic to $W$ . 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\}

.

Another copy of $W$ 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\}

.

So can the third:

W_{3}=\{ax_{1}^{3}+bx_{2}^{3}+cx_{3}^{3}|a+b+c=0\}

.

Then $W_{1}\oplus W_{2}\oplus W_{3}$ is the isotypic component of type $W$ in $V$ .

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,[21] 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\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.[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 $S^{1}$ , the theorem exactly amounts to the classical Fourier analysis.[23]

Applications to physics

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]

Notes

To be precise, the theorem concerns the regular representation of $G\times G$ and the above statement is a corollary.

References

Citations

Procesi 2007, Ch. 6, § 1.1, Definition 1 (ii).
Procesi 2007, Ch. 6, § 2.1.
Anderson & Fuller 1992, Proposition 9.4.
Anderson & Fuller 1992, Theorem 9.6.
Anderson & Fuller 1992, Lemma 9.2.
Fulton & Harris 1991, § 9.3. A
Hall 2015, Theorem 4.28
Fulton & Harris 1991, Corollary 1.6.
Serre 1977, Theorem 2.
Hall 2015 Theorem 10.9
Jacobson 1989, § 3.5. Exercise 4.
Artin 1999, Ch. V, § 14.
Fulton & Harris 1991, just after Corollary 1.6.
Serre 1977, § 1.4. remark
Procesi 2007, Ch. 6, § 2.3.
Fulton & Harris 1991, Proposition 1.8.
Fulton & Harris 1991, § 2.3.
Fulton & Harris 1991, § 2.1. Definition
Serre 1977, § 2.3. Theorem 3 and § 4.3.
Serre 1977, § 2.6. Theorem 8 (i)
Procesi 2007, Ch. 8, Theorem 3.2.
Serre 1977, § 2.4. Corollary 1 to Proposition 5
Procesi 2007, Ch. 8, § 3.3.
Hall, Brian C. (2013). "Angular Momentum and Spin". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. 267. Springer. pp. 367–392. ISBN 978-1461471158.
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. doi:10.1063/1.524268.

Sources

Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, 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; 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).
Artin, Michael (1999). "Noncommutative Rings" (PDF).CS1 maint: ref=harv (link)
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. 222 (2nd ed.). Springer. ISBN 978-3319134666.CS1 maint: ref=harv (link)
Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5
Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 9780387260402.CS1 maint: ref=harv (link).
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.CS1 maint: ref=harv (link)

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[22] To be precise, the theorem concerns the regular representation of $G\times G$ and the above statement is a corollary.

[Procesi-1] Procesi 2007, Ch. 6, § 1.1, Definition 1 (ii).

[2] Procesi 2007, Ch. 6, § 2.1.

[3] Anderson & Fuller 1992, Proposition 9.4.

[4] Anderson & Fuller 1992, Theorem 9.6.

[5] Anderson & Fuller 1992, Lemma 9.2.

[Fulton-6] Fulton & Harris 1991, § 9.3. A

[7] Hall 2015, Theorem 4.28

[FOOTNOTEFultonHarris1991Corollary_1.6-8] Fulton & Harris 1991, Corollary 1.6.

[FOOTNOTESerre1977Theorem_2-9] Serre 1977, Theorem 2.

[10] Hall 2015 Theorem 10.9

[11] Jacobson 1989, § 3.5. Exercise 4.

[12] Artin 1999, Ch. V, § 14.

[13] Fulton & Harris 1991, just after Corollary 1.6.

[14] Serre 1977, § 1.4. remark

[isotypic-15] Procesi 2007, Ch. 6, § 2.3.

[Fulton_Prop_1.8.-16] Fulton & Harris 1991, Proposition 1.8.

[17] Fulton & Harris 1991, § 2.3.

[18] Fulton & Harris 1991, § 2.1. Definition

[19] Serre 1977, § 2.3. Theorem 3 and § 4.3.

[20] Serre 1977, § 2.6. Theorem 8 (i)

[21] Procesi 2007, Ch. 8, Theorem 3.2.

[23] Serre 1977, § 2.4. Corollary 1 to Proposition 5

[24] Procesi 2007, Ch. 8, § 3.3.

[Hall-17-25] Hall, Brian C. (2013). "Angular Momentum and Spin". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. 267. Springer. pp. 367–392. ISBN 978-1461471158.

[26] 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. doi:10.1063/1.524268.