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
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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
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* ''Simple'' (album), by Andy Yorke, 2008, and its title track
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(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
.
The equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:
''Proof of the lemma'': Write
where
are simple representations. Without loss of generality, we can assume
are subrepresentations; i.e., we can assume the direct sum is internal. Now, consider the family of all possible direct sums
with various subsets
. Put the partial ordering on it by saying the direct sum over ''K'' is less than the direct sum over ''J'' if
. By
Zorn's lemma, we can find a maximal
such that
. We claim that
. By definition,
so we only need to show that
. If
is a proper subrepresentatiom of
then there exists
such that
. Since
is simple (irreducible),
. This contradicts the maximality of
, so
as claimed. Hence,
is a section of ''p''.
Note that we cannot take
to the set of
such that
. The reason is that it can happen, and frequently does, that
is a subspace of
and yet
. For example, take
,
and
to be three distinct lines through the origin in
. For an explicit counterexample, let
be the algebra of
matrices and set
, the regular representation of
. Set
and
and set
. Then
,
and
are all irreducible
-modules and
. Let
be the natural surjection. Then
and
. In this case,
but
because this sum is not direct.
''Proof of equivalences''
: Take ''p'' to be the natural surjection
. Since ''V'' is semisimple, ''p'' splits and so, through a section,
is isomorphic to a subrepretation that is complementary to ''W''.
: 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.,
for some
. By modular law, it implies
. Then
is a simple subrepresentation
of ''W'' ("simple" because of maximality). This establishes the observation. Now, take
to be the sum of all simple subrepresentations, which, by 3., admits a complementary representation
. If
, then, by the early observation,
contains a simple subrepresentation and so
, a nonsense. Hence,
.
: 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
is a sum of simple subrepresentations, a semisimple decomposition
, some subset
, can be extracted from the sum.
As in the proof of the lemma, we can find a maximal direct sum
that consists of some
’s. Now, for each ''i'' in ''I'', by simplicity, either
or
. In the second case, the direct sum
is a contradiction to the maximality of ''W''. Hence,
.
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
and
, then
for any ''w'' in ''W'' since ''W'' is ''G''-invariant, and so
.
For example, given a continuous finite-dimensional complex representation
of a finite group or a compact group ''G'', by the averaging argument, one can define an
inner product on ''V'' that is ''G''-invariant: i.e.,
, which is to say
is a unitary operator and so
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:
such that each successive quotient
is a simple representation. Then the associated vector space
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
to be the group consisting of real matrices
; it acts on
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
. 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):
:
where are simple representations, mutually non-isomorphic to one another, and are positive integers. By Schur's lemma,
:,
where refers to the equivariant linear maps. Also, each is unchanged if is replaced by another simple representation isomorphic to . Thus, the integers are independent of chosen decompositions; they are the ''multiplicities'' of simple representations , up to isomorphisms, in ''V''.
In general, given a finite-dimensional representation of a group ''G'' over a field ''k'', the composition is called the character of . When is semisimple with the decomposition as above, the trace is the sum of the traces of with multiplicities and thus, as functions on ''G'',
:
where are the characters of . When ''G'' is a finite group or more generally a compact group and 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 .
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:
:
where is the set of isomorphism classes of simple representations of ''G'' and is the isotypic component of ''V'' of type ''S'' for some .
Example
Let be the space of homogeneous degree-three polynomials over the complex numbers in variables . Then acts on 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 . In particular, contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representation of . For example, the span of and is isomorphic to . This can more easily be seen by writing this two-dimensional subspace as
:.
Another copy of can be written in a similar form:
:.
So can the third:
:.
Then is the isotypic component of type in .
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 = the Hilbert space of (classes of) square-integrable functions on a compact group ''G'':
:
where means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations 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