In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Schur's lemma is an elementary but extremely useful statement in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
groups
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 ...
and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
. In the group case it says that if ''M'' and ''N'' are two finite-dimensional
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s
of a group ''G'' and ''φ'' is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
from ''M'' to ''N'' that commutes with the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the group, then either ''φ'' is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, or ''φ'' = 0. An important special case occurs when ''M'' = ''N'', i.e. ''φ'' is a self-map; in particular, any element of the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of a group must act as a scalar operator (a scalar multiple of the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
) on ''M''. The
lemma is named after
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the ...
who used it to
prove 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 ...
and develop the basics of the
representation theory of finite groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
. Schur's lemma admits generalisations to
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, the most common of which are due to
Jacques Dixmier
Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping.
Biogra ...
and
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
.
Representation theory of groups
Representation theory is the study of
homomorphisms from a group, ''G'', into the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
''GL''(''V'') of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V''; i.e., into the
group of automorphisms of ''V''. (Let us here restrict ourselves to the case when the underlying
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of ''V'' is
, the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s.) Such a homomorphism is called a representation of ''G'' on ''V''. A representation on ''V'' is a special case of a
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
on ''V'', but rather than permit any arbitrary
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s (
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s) of the underlying set of ''V'', we restrict ourselves to invertible ''linear'' transformations.
Let ''ρ'' be a representation of ''G'' on ''V''. It may be the case that ''V'' has a
subspace, ''W'', such that for every element ''g'' of ''G'', the invertible linear map ''ρ''(''g'') preserves or fixes ''W'', so that (''ρ''(''g''))(''w'') is in ''W'' for every ''w'' in ''W'', and (''ρ''(''g''))(''v'') is not in ''W'' for any ''v'' not in ''W''. In other words, every linear map ''ρ''(''g''): ''V''→''V'' is also an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of ''W'', ''ρ''(''g''): ''W''→''W'', when its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is restricted to ''W''. We say ''W'' is stable under ''G'', or stable under the action of ''G''. It is clear that if we consider ''W'' on its own as a vector space, then there is an obvious representation of ''G'' on ''W''—the representation we get by restricting each map ''ρ''(''g'') to ''W''. When ''W'' has this property, we call ''W'' with the given representation a ''subrepresentation'' of ''V''. Every representation of ''G'' has itself and the
zero vector space
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforeme ...
as trivial subrepresentations. A representation of ''G'' with no non-trivial subrepresentations is called an ''
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
''. Irreducible representations – like the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, or like the
simple groups in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
– are the building blocks of representation theory. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations.
As we are interested in homomorphisms between groups, or
continuous maps
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
between
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s, we are interested in certain functions between representations of ''G''. Let ''V'' and ''W'' be vector spaces, and let
and
be representations of ''G'' on ''V'' and ''W'' respectively. Then we define a map ''f'' from ''V'' to ''W'' to be a linear map from ''V'' to ''W'' that is
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
under the action of ''G''; that is, for every ''g'' in ''G'',
. In other words, we require that ''f'' commutes with the action of ''G''. maps are the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s in the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of representations of ''G''.
Schur's Lemma is a theorem that describes what maps can exist between two irreducible representations of ''G''.
Statement and Proof of the Lemma
Theorem ''(Schur's Lemma)'': Let ''V'' and ''W'' be vector spaces; and let
and
be irreducible representations of ''G'' on ''V'' and ''W'' respectively.
# If
and
are not
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
, then there are no nontrivial maps between them.
# If
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
(e.g.
); and if
, then the only nontrivial maps are the identity, and scalar multiples of the identity. (A scalar multiple of the identity is sometimes called a ''homothety.'')
Proof: Suppose
is a nonzero map from
to
. We will prove that
and
are isomorphic. Let
be the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
, or null space, of
in
, the subspace of all
in
for which
. (It is easy to check that this is a subspace.) By the assumption that
is , for every
in
and choice of
in
. But saying that
is the same as saying that
is in the null space of
. So
is stable under the action of ''G''; it is a subrepresentation. Since by assumption
is irreducible,
must be zero; so
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
.
By an identical argument we will show
is also
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
; since
, we can conclude that for arbitrary choice of
in the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of
,
sends
somewhere else in the image of
; in particular it sends it to the image of
. So the image of
is a subspace
of
stable under the action of
, so it is a subrepresentation and
must be zero or surjective. By assumption it is not zero, so it is surjective, in which case it is an isomorphism.
In the event that
finite-dimensional over an algebraically closed field and they have the same representation, let
be an
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
. (An eigenvalue exists for every linear transformation on a finite-dimensional vector space over an algebraically closed field.) Let
. Then if
is an
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
corresponding to
. It is clear that
is a map, because the sum or difference of maps is also . Then we return to the above argument, where we used the fact that a map was to conclude that the kernel is a subrepresentation, and is thus either zero or equal to all of
; because it is not zero (it contains
) it must be all of ''V'' and so
is trivial, so
.
Corollary of Schur's Lemma
An important corollary of Schur's lemma follows from the observation that we can often build explicitly
-linear maps between representations by "averaging" over the action of individual group elements on some fixed linear operator. In particular, given any irreducible representation, such objects will satisfy the assumptions of Schur's lemma, hence be scalar multiples of the identity. More precisely:
Corollary: Using the same notation from the previous theorem, let
be a linear mapping of ''V'' into ''W'', and set
Then,
# If
and
are not
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
, then
.
# If
is
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
(e.g.
); and if
, then
, where ''n'' is the dimension of ''V''. That is,
is a homothety of ratio
.
Proof:
Let us first show that
is a G-linear map, i.e.,
for all
. Indeed, consider that
Now applying the previous theorem, for case 1, it follows that
, and for case 2, it follows that
is a scalar multiple of the identity matrix (i.e.,
). To determine the scalar multiple
, consider that
It then follows that
.
This result has numerous applications. For example, in the context of
quantum information science
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
, it is used to derive results about
complex projective t-designs.
Formulation in the language of modules
If ''M'' and ''N'' are two
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cycl ...
s over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'', then any
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
''f'': ''M'' → ''N'' of ''R''-
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
is either invertible or zero. In particular, the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
of a simple module is a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
.
The condition that ''f'' is a module homomorphism means that
:
The group version is a special case of the module version, since any representation of a group ''G'' can equivalently be viewed as a 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 give ...
of ''G''.
Schur's lemma is frequently applied in the following particular case. Suppose that ''R'' is 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 ...
over a field ''k'' and the vector space ''M'' = ''N'' is a simple module of ''R''. Then Schur's lemma says that the endomorphism ring of the module ''M'' is a
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
over ''k''. If ''M'' is finite-dimensional, this division algebra is finite-dimensional. If ''k'' is the field of complex numbers, the only option is that this division algebra is the complex numbers. Thus the endomorphism ring of the module ''M'' is "as small as possible". In other words, the only linear transformations of ''M'' that commute with all transformations coming from ''R'' are scalar multiples of the identity.
This holds more generally for any algebra
over an
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
algebraically closed field
and for any simple module
that is at most countably-dimensional: the only linear transformations of
that commute with all transformations coming from
are scalar multiples of the identity.
When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over a
-algebra is said to be
absolutely simple if its endomorphism ring is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to
. This is in general stronger than being irreducible over the field
, and implies the module is irreducible even over the
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...
of
.
Representations of Lie groups and Lie algebras
We now describe Schur's lemma as it is usually stated in the context of representations of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s. There are three parts to the result.
First, suppose that
and
are irreducible representations of a Lie group or Lie algebra over any field and that
is an
intertwining map. Then
is either zero or an isomorphism.
Second, if
is an irreducible representation of a Lie group or Lie algebra over an ''algebraically closed'' field and
is an intertwining map, then
is a scalar multiple of the identity map.
Third, suppose
and
are irreducible representations of a Lie group or Lie algebra over an ''algebraically closed'' field and
are nonzero
intertwining maps. Then
for some scalar
.
A simple
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of the second statement is that every complex irreducible representation of an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
is one-dimensional.
Application to the Casimir element
Suppose
is a Lie algebra and
is the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
of
. Let
be an irreducible representation of
over an algebraically closed field. The
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of the universal enveloping algebra ensures that
extends to a representation of
acting on the same vector space. It follows from the second part of Schur's lemma that if
belongs to the center of
, then
must be a multiple of the identity operator. In the case when
is a complex
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 i ...
, an important example of the preceding construction is the one in which
is the (quadratic)
Casimir element . In this case,
, where
is a constant that can be computed explicitly in terms of the highest weight of
. The action of the Casimir element plays an important role in the proof of complete reducibility for finite-dimensional representations of semisimple Lie algebras.
[ Section 10.3]
See also
Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.
Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...
.
Generalization to non-simple modules
The one module version of Schur's lemma admits generalizations involving modules ''M'' that are not necessarily simple. They express relations between the module-theoretic properties of ''M'' and the properties of the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
of ''M''.
A module is said to be strongly indecomposable if its endomorphism ring is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
. For the important class of modules of
finite length, the following properties are equivalent :
* A module ''M'' is
indecomposable;
* ''M'' is strongly indecomposable;
* Every endomorphism of ''M'' is either nilpotent or invertible.
In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
. However, even over the ring of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
, the module of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s has an endomorphism ring that is a division ring, specifically the field of rational numbers. Even for group rings, there are examples when the
characteristic of the field divides the
order of the group: the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yie ...
of the
projective cover of the one-dimensional representation of the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
A
5 over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with three elements F
3 has F
3 as its endomorphism ring.
See also
*
Quillen's lemma
Notes
References
*
*
*
*
* {{SpringerEOM, title=Schur lemma, id=Schur_lemma&oldid=17919, author-last1=Shtern, author-last2=Lomonosov, author-first1=A.I., author-first2=V.I.
Representation theory
Lemmas