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In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
or an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications 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 In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
''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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
(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: # ''V'' is semisimple as a representation. # ''V'' is a sum of simple
subrepresentation In representation theory, a subrepresentation of a representation (\pi, V) of a group ''G'' is a representation (\pi, _W, W) such that ''W'' is a vector subspace In mathematics, and more specifically in linear algebra, a linear subspace, also kn ...
s. # 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: ''Proof of the lemma'': Write V = \bigoplus_ 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 := \bigoplus_ 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. By Zorn's lemma, we can find a maximal J \subset I such that \operatornamep\cap V_J=0. We claim that V=\operatornamep\oplus V_J. By definition, \operatornamep\cap V_J=0 so we only need to show that V=\operatornamep+V_J. If \operatornamep+V_J is a proper subrepresentatiom of V then there exists k\in I - J such that V_k\not\subset \operatornamep+V_J. Since V_k is simple (irreducible), V_k\cap(\operatornamep+V_J)=0. This contradicts the maximality of J, so V=\operatornamep\oplus V_J as claimed. Hence, W \simeq V/\operatornamep\simeq V_J \to V is a section of ''p''. \square Note that we cannot take J to the set of i such that \ker(p) \cap V_i = 0. The reason is that it can happen, and frequently does, that X is a subspace of Y\oplus Z and yet X\cap Y=0=X\cap Z. For example, take X, Y and Z to be three distinct lines through the origin in \mathbb^2. For an explicit counterexample, let A=\operatorname_2(F) be the algebra of 2\times2 matrices and set V=A, the regular representation of A. Set V_1=\Bigl\ and V_2=\Bigl\ and set W=\Bigl\. Then V_1, V_2 and W are all irreducible A-modules and V=V_1\oplus V_2. Let p: V\to V/W be the natural surjection. Then \operatornamep=W\ne0 and V_1\cap\operatornamep=0=V_2\cap\operatornamep. In this case, W\simeq V_1\simeq V_2 but V\ne\operatornamep\oplus V_1\oplus V_2 because this sum is not direct. ''Proof of equivalences'' 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' \ne 0, then, by the early observation, W' contains a simple subrepresentation and so W \cap W' \ne 0, a nonsense. Hence, W' = 0. 2. \Rightarrow 1.: 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 = \sum_ V_i is a sum of simple subrepresentations, a semisimple decomposition V = \bigoplus_ V_i, some subset I' \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 \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 In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
(i.e., a representation factoring through a unitary group) is a basic example of a semisimple representation. Such a representation is semisimple since if ''W'' is a subrepresentation, then the orthogonal complement to ''W'' is a complementary representation because if v \in W^ and g \in G, then \langle \pi(g) v, w \rangle = \langle v, \pi(g^) w \rangle = 0 for any ''w'' in ''W'' since ''W'' is ''G''-invariant, and so \pi(g) v \in W^. For example, given a continuous finite-dimensional complex representation \pi: G \to GL(V) of a finite group or a compact group ''G'', by the averaging argument, one can define an inner product \langle, \rangle on ''V'' that is ''G''-invariant: i.e., \langle \pi(g) v, \pi(g) w \rangle = \langle v, w \rangle, which is to say \pi(g) is a unitary operator and so \pi is a unitary representation. Hence, every finite-dimensional continuous complex representation of ''G'' is semisimple. For a finite group, this is a special case of Maschke's theorem, which says a finite-dimensional representation of a finite group ''G'' over a field k with characteristic not dividing the order of ''G'' is semisimple.


Representations of semisimple Lie algebras

By
Weyl's theorem on complete reducibility In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let \mathfrak be a semisimple Lie algebra over a field o ...
, every finite-dimensional representation of a
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
over a field of characteristic zero is semisimple.


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.


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_ is a simple representation. Then the associated vector space \operatorname V := \bigoplus_^V_i/V_ is a semisimple representation called an associated semisimple representation, which, up to an isomorphism, is uniquely determined by ''V''.


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 1 & a \\ & 1 \end; it acts on V = \mathbb^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 1 \\ 0 \end. That is, there are exactly three ''G''-subrepresentations of ''V''; in particular, ''V'' is not semisimple (as a unique one-dimensional subrepresentation does not admit a complementary representation).


Semisimple decomposition and multiplicity

The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example, for a trivial representation, simple representations are one-dimensional vector spaces and thus a semisimple decomposition amounts to a choice of a basis of the representation vector space. The isotypic decomposition, on the other hand, is an example of a unique decomposition. However, for a finite-dimensional semisimple representation ''V'' over an algebraically closed field, the numbers of simple representations up to isomorphisms appearing in the decomposition of ''V'' (1) are unique and (2) completely determine the representation up to isomorphisms; this is a consequence of Schur's lemma in the following way. Suppose a finite-dimensional semisimple representation ''V'' over an algebraically closed field is given: by definition, it is a direct sum of simple representations. By grouping together simple representations in the decomposition that are isomorphic to each other, up to an isomorphism, one finds a decomposition (not necessarily unique): :V \simeq \bigoplus_i V_i^ 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_(V_i, V) = \dim \operatorname_(V, V_i), where \operatorname_ refers to the equivariant linear maps. Also, each m_i is unchanged if V_i is replaced by another simple representation isomorphic to V_i. Thus, the integers m_i are independent of chosen decompositions; they are the ''multiplicities'' of simple representations V_i, up to isomorphisms, in ''V''. In general, given a finite-dimensional representation \pi: G \to GL(V) of a group ''G'' over a field ''k'', the composition \chi_V : G \overset\to GL(V) \overset \to k is called the character of (\pi, V). When (\pi, V) is semisimple with the decomposition V \simeq \bigoplus_i V_i^ as above, the trace \operatorname(\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_ where \chi_ 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 In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in ...
say: the irreducible characters (characters of simple representations) of ''G'' are an orthonormal subset of the space of complex-valued functions on ''G'' and thus m_i = \langle \chi_V, \chi_ \rangle.


Isotypic decomposition

There is a decomposition of a semisimple representation that is unique, called ''the'' isotypic decomposition of the representation. By definition, given a simple representation ''S'', the isotypic component of type ''S'' of a representation ''V'' is the sum of all subrepresentations of ''V'' that are isomorphic to ''S''; note the component is also isomorphic to the direct sum of some choice of subrepresentations isomorphic to ''S'' (so the component is unique, while the summands are not necessary so). Then the isotypic decomposition of a semisimple representation ''V'' is the (unique) direct sum decomposition: :V = \bigoplus_ V^ where \widehat is the set of isomorphism classes of simple representations of ''G'' and V^ 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_1^2x_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=\. Another copy of W can be written in a similar form: : W_2=\. So can the third: : W_3=\. 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 In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Pete ...
, 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, 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 V^ where \widehat 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 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 /math> is simply the group algebra of ''G'' and also the completion is vacuous. Thus, the theorem simply says that : \mathbb = \bigoplus_ V^. That is, each simple representation of ''G'' appears in the regular representation with multiplicity the dimension of the representation. This is one of standard facts in the representation theory of a finite group (and is much easier to prove). When the group ''G'' is the
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
S^1, the theorem exactly amounts to the classical Fourier analysis.


Applications to physics

In quantum mechanics and particle physics, the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
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 Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
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. In
angular momentum coupling In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact t ...
,
Clebsch–Gordan coefficients In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
arise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.


Notes


References


Citations


Sources

* ; 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). * * * * * . * {{refend Representation theory