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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 \mathbb, 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 \rho_V and \rho_W 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'', \rho_W(g) \circ f = f \circ \rho_V(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 \rho_V and \rho_W be irreducible representations of ''G'' on ''V'' and ''W'' respectively. # If V and W 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 V=W finite-dimensional over an algebraically closed field (e.g. \mathbb); and if \rho_V = \rho_W, 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 f is a nonzero map from V to W. We will prove that V and W are isomorphic. Let V' be the kernel, or null space, of f in V, described as V' = \. V' is a subspace of V as it is nonempty (contains the zero vector), and is closed. By the assumption that f is , for every g in G and choice of x in V', f((\rho_V(g))(x)) = (\rho_W(g))(f(x))=(\rho_W(g))(0) = 0. But saying that f(\rho_V(g)(x))=0 is the same as saying that \rho_V(g)(x) is in the null space of f:V\rightarrow W. So V' is stable under the action of ''G''; it is a subrepresentation. Since by assumption V is irreducible, V' must be zero; so f is injective. By an identical argument we will show f is also surjective; since f((\rho_V(g))(x)) = (\rho_W(g))(f(x)), we can conclude that for arbitrary choice of f(x) 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 f, \rho_W(g) sends f(x) somewhere else in the image of f; in particular it sends it to the image of \rho_V(g)x. So the image of f(x) is a subspace W' of W stable under the action of G, so it is a subrepresentation and f 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 V=W finite-dimensional over an algebraically closed field and they have the same representation, let \lambda be an eigenvalue of f. (An eigenvalue exists for every linear transformation on a finite-dimensional vector space over an algebraically closed field.) Let f' = f-\lambda I. Then if x is an eigenvector of f corresponding to \lambda, f'(x)=0. It is clear that f' 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 V; because it is not zero (it contains x) it must be all of ''V'' and so f' is trivial, so f = \lambda I.


Corollary of Schur's Lemma

An important corollary of Schur's lemma follows from the observation that we can often build explicitly G-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 h be a linear mapping of ''V'' into ''W'', and set h_0 = \frac \sum_ (\rho_W(g))^ h \rho_V(g). Then, # If V and W 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 h_0 = 0. # If V=W is finite-dimensional over an algebraically closed field (e.g. \mathbb); and if \rho_V = \rho_W, then h_0 = I\, \mathrm n, where ''n'' is the dimension of ''V''. That is, h_0 is a homothety of ratio \mathrm n. Proof: Let us first show that h_0 is a G-linear map, i.e., \rho_W(g) \circ h_0 = h_0 \circ \rho_V(g) for all g \in G. Indeed, consider that \begin (\rho_W(g'))^ h_0 \rho_V(g') & = \frac \sum_ (\rho_W(g'))^(\rho_W(g))^ h \rho_V(g) \rho_V(g') \\ & = \frac \sum_ (\rho_W(g\circ g'))^ h \rho_V(g\circ g') \\ & = h_0 \end Now applying the previous theorem, for case 1, it follows that h_0 = 0, and for case 2, it follows that h_0 is a scalar multiple of the identity matrix (i.e., h_0 = \mu I). To determine the scalar multiple \mu, consider that \mathrm _0= \frac \sum_ \mathrm \rho_V(g))^ h \rho_V(g) = \mathrm h It then follows that \mu = \mathrm / n. 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 M,N be two simple modules over a ring R. 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 ...
f\colon M \to N of R- 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 f: since \ker(f) \subseteq M, \mathrm(f) \subseteq N are submodules of simple modules, by definition they are either zero or equal to M,N respectively. In particular, we have that either \ker(f)=M, meaning that f is the zero morphism, or that \ker(f)=0, meaning that f is injective. In the latter case, the first isomorphism theorem tells us furthermore that \mathrm(f) \cong M/\ker(f)\cong M is not trivial and thus \mathrm(f)=N: this shows that f is in addition surjective, hence bijective and thus an isomorphism of R-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 R is an algebra over an algebraically closed field k and M is a simple R-module satisfying \dim_k(M) < \# k (the cardinality of k), then \operatorname_R(M) = k. So in particular, if R is an algebra over an uncountable algebraically closed field k and M is a simple module that is at most countably-dimensional, the only linear transformations of M that commute with all transformations coming from R 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 k-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 k. This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.


Application to central characters

Definition: Let R be a k-algebra. An R-module M is said to have ''central character'' \chi: Z(R)\to k (here, Z(R) is the center of R) if for every m\in M, z\in Z(R) there is n\in\N such that (z-\chi(z))^nm=0, i.e. if every m\in M is a generalized eigenvector of z with eigenvalue \chi(z). If \operatorname_R(M)=k, say in the case sketched above, every element of Z(R) acts on M as an R-endomorphism and hence as a scalar. Thus, there is a ring homomorphism \chi: Z(R)\to k such that (z-\chi(z))m =0 for all z\in Z(R), m\in M. In particular, M has central character \chi. If R=U(\mathfrak), k=\C 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 \mathfrak is finite-dimensional (so that R=U(\mathfrak) is countable-dimensional), then every simple \mathfrak-module has an infinitesimal character. In the case where k=\C, R=\C /math> is the group algebra of a finite group G, the same conclusion follows. Here, the center of R consists of elements of the shape \sum_ a(g) g where a: G\to\C is a class function, i.e. invariant under conjugation. Since the set of class functions is spanned by the characters \chi_\pi of the irreducible representations \pi\in\hat, the central character is determined by what it maps u_\pi:= \frac\sum_ \chi_\pi(g) g to (for all \pi\in\hat). Since all u_\pi are idempotent, they are each mapped either to 0 or to 1, and since u_\pi u_=0 for two different irreducible representations, only one u_\pi can be mapped to 1: the one corresponding to the module M.


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 (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. There are three parts to the result. First, suppose that V_1 and V_2 are irreducible representations of a Lie group or Lie algebra over any field and that \phi:V_1\rightarrow V_2 is an intertwining map. Then \phi is either zero or an isomorphism. Second, if V is an irreducible representation of a Lie group or Lie algebra over an ''algebraically closed'' field and \phi:V\rightarrow V is an intertwining map, then \phi is a scalar multiple of the identity map. Third, suppose V_1 and V_2 are irreducible representations of a Lie group or Lie algebra over an ''algebraically closed'' field and \phi_1, \phi_2:V_1\rightarrow V_2 are nonzero intertwining maps. Then \phi_1=\lambda\phi_2 for some scalar \lambda. A simple corollary of the second statement is that every complex irreducible representation of an abelian group is one-dimensional.


Application to the Casimir element

Suppose \mathfrak is a Lie algebra and U(\mathfrak) is the universal enveloping algebra of \mathfrak. Let \pi:\mathfrak\rightarrow\mathrm(V) be an irreducible representation of \mathfrak over an algebraically closed field. The universal property of the universal enveloping algebra ensures that \pi extends to a representation of U(\mathfrak) acting on the same vector space. It follows from the second part of Schur's lemma that if x belongs to the center of U(\mathfrak), then \pi(x) must be a multiple of the identity operator. In the case when \mathfrak is a complex
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
, an important example of the preceding construction is the one in which x is the (quadratic) Casimir element C. In this case, \pi(C)=\lambda_\pi I, where \lambda_\pi is a constant that can be computed explicitly in terms of the highest weight of \pi. 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


Generalization to non-simple modules

The one-module version of Schur's lemma admits generalizations for modules M that are not necessarily simple. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M. Theorem : A module is said to be strongly indecomposable if its endomorphism ring is a
local ring In mathematics, more specifically in 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 algebraic varieties or manifolds, or of ...
. For a module M of finite length, the following properties are equivalent: * M is indecomposable; * M is strongly indecomposable; * Every endomorphism of M is either nilpotent or invertible. Schur's lemma cannot be reversed in general, however, since there exist modules that are not simple but whose endomorphism algebra is a division ring. Such modules are necessarily indecomposable and so cannot exist over semisimple rings, such as the complex group ring of a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
. However, even over the ring of
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, 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 (for example, The set of all ...
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 of the projective cover of the one-dimensional representation of the
alternating group In mathematics, an alternating group is the Group (mathematics), 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 ...
A5 over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
with three elements F3 has F3 as its endomorphism ring.


See also

* Schur complement * 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 Theorems in representation theory Issai Schur