In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Schur's lemma is an elementary but extremely useful statement in
representation theory of
groups and
algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional
irreducible representations
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 mapping V \to W between two vector spaces that p ...
from ''M'' to ''N'' that commutes with the
action of the group, then either ''φ'' is
invertible, 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 of a group must act as a scalar operator (a scalar multiple of the
identity) 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 Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
who used it to
prove the
Schur orthogonality relations and develop the basics of the
representation theory of finite groups. Schur's lemma admits generalisations to
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s, the most common of which are due to
Jacques Dixmier and
Daniel Quillen.
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 n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
''GL''(''V'') of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V''; i.e., into the
group of automorphisms of ''V''. (Let us here restrict ourselves to the case when the underlying
field 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 for ...
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 of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
on ''V'', but rather than permit any arbitrary
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
s (
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
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 of ''W'', ''ρ''(''g''): ''W''→''W'', when its
domain 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 as trivial subrepresentations. A representation of ''G'' with no non-trivial subrepresentations is called an ''
irreducible representation''. Irreducible representations – like the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
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.
Just as we are interested in homomorphisms between groups, and in
continuous maps between
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, we are also 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 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
morphisms in the
category 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 or morphism 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 the ...
, then there are no nontrivial maps between them.
# If
finite-dimensional over an
algebraically closed field (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, or null space, of
in
, described as
.
is a subspace of
as it is nonempty (contains the zero vector), and is closed.
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.
By an identical argument we will show
is also
surjective; since
, we can conclude that for arbitrary choice of
in the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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 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 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 or morphism 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 the ...
, then
.
# If
is
finite-dimensional over an
algebraically closed field (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, it is used to derive results about
complex projective t-designs. In the context of
molecular orbital theory, it is used to restrict atomic orbital interactions based on the
molecular symmetry.
Formulation in the language of modules
Theorem: Let
be two
simple modules over a
ring . Then any
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of
-
modules is either zero or an isomorphism. In particular, the
endomorphism ring of a simple module is a
division ring.
Proof: Consider the
kernel and
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of
: since
are submodules of simple modules, by definition they are either zero or equal to
respectively. In particular, we have that either
, meaning that
is the zero morphism, or that
, meaning that
is injective. In the latter case, the
first isomorphism theorem tells us furthermore that
is not trivial and thus
: this shows that
is in addition surjective, hence bijective and thus an isomorphism of
-modules.
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 of ''G''.
Schur's lemma is frequently applied in the following particular case. Suppose that ''R'' is an
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
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 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.
More generally, if
is an algebra over an algebraically closed field
and
is a simple
-module satisfying
(the cardinality of
), then
. So in particular, if
is an algebra over an
uncountable algebraically closed field
and
is a 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 or morphism 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 the ...
to
. This is in general stronger than being irreducible over the field
, and implies the module is irreducible even over the
algebraic closure of
.
Application to central characters
Definition: Let
be a
-algebra. An
-module
is said to have ''central character''
(here,
is the
center of
) if for every
there is
such that
, i.e. if every
is a
generalized eigenvector of
with eigenvalue
.
If
, say in the case sketched above, every element of
acts on
as an
-endomorphism and hence as a scalar. Thus, there is a ring homomorphism
such that
for all
. In particular,
has central character
.
If
is the
universal enveloping algebra of a Lie algebra, a central character is also referred to as an
infinitesimal character and the previous considerations show that if
is finite-dimensional (so that
is countable-dimensional), then every simple
-module has an infinitesimal character.
In the case where