Schur's lemma

Schur's lemma For other uses, see Schur's lemma (disambiguation).

In mathematics, Schur's lemma[1] 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 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 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 groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.

Contents 1 Representation theory of groups 1.1 Statement and Proof of the Lemma 1.2 Corollary of Schur's Lemma 2 Formulation in the language of modules 3 Representations of Lie groups and Lie algebras 3.1 Application to the Casimir element 4 Generalization to non-simple modules 5 See also 6 Notes 7 References Representation theory of groups Representation theory is the study of homomorphisms from a group, G, into the general linear group GL(V) of a vector space 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 {displaystyle mathbb {C} } , the field of complex numbers.) Such a homomorphism is called a representation of G on V. A representation on V is a special case of a group action on V, but rather than permit any arbitrary bijections (permutations) 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 numbers, or like the simple groups in group theory – 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 between topological spaces, we are interested in certain functions between representations of G. Let V and W be vector spaces, and let {displaystyle rho _{V}} and {displaystyle rho _{W}} be representations of G on V and W respectively. Then we define a G-linear 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, {displaystyle rho _{W}(g)circ f=fcirc rho _{V}(g)} . In other words, we require that f commutes with the action of G. G-linear maps are the morphisms in the category of representations of G.

Schur's Lemma is a theorem that describes what G-linear 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 {displaystyle rho _{V}} and {displaystyle rho _{W}} be irreducible representations of G on V and W respectively.[2] If {displaystyle V} and {displaystyle W} are not isomorphic, then there are no nontrivial G-linear maps between them. If {displaystyle V=W} finite-dimensional over an algebraically closed field (e.g. {displaystyle mathbb {C} } ); and if {displaystyle rho _{V}=rho _{W}} , then the only nontrivial G-linear maps are the identity, and scalar multiples of the identity. (A scalar multiple of the identity is sometimes called a homothety.) Proof: Suppose {displaystyle f} is a nonzero G-linear map from {displaystyle V} to {displaystyle W} . We will prove that {displaystyle V} and {displaystyle W} are isomorphic. Let {displaystyle V'} be the kernel, or null space, of {displaystyle f} in {displaystyle V} , the subspace of all {displaystyle x} in {displaystyle V} for which {displaystyle fx=0} . (It is easy to check that this is a subspace.) By the assumption that {displaystyle f} is G-linear, for every {displaystyle g} in {displaystyle G} and choice of {displaystyle x} in {displaystyle V',f((rho _{V}(g)(x))=(rho _{W}(g))(f(x))=(rho _{W}(g))(0)=0} . But saying that {displaystyle f(rho _{V}(g)(x))=0} is the same as saying that {displaystyle rho _{V}(g)(x)} is in the null space of {displaystyle f:Vrightarrow W} . So {displaystyle V'} is stable under the action of G; it is a subrepresentation. Since by assumption {displaystyle V} is irreducible, {displaystyle V'} must be zero; so {displaystyle f} is injective.

By an identical argument we will show {displaystyle f} is also surjective; since {displaystyle f((rho _{V}(g))(x))=(rho _{W}(g))(f(x))} , we can conclude that for arbitrary choice of {displaystyle f(x)} in the image of {displaystyle f} , {displaystyle rho _{W}(g)} sends {displaystyle f(x)} somewhere else in the image of {displaystyle f} ; in particular it sends it to the image of {displaystyle rho _{V}(g)x} . So the image of {displaystyle f(x)} is a subspace {displaystyle W'} of {displaystyle W} stable under the action of {displaystyle G} , so it is a subrepresentation and {displaystyle 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 {displaystyle V=W} finite-dimensional over an algebraically closed field and they have the same representation, let {displaystyle lambda } be an eigenvalue of {displaystyle f} . (An eigenvalue exists for every linear transformation on a finite-dimensional vector space over an algebraically closed field.) Let {displaystyle f'=f-lambda I} . Then if {displaystyle x} is an eigenvector of {displaystyle f} corresponding to {displaystyle lambda ,f'(x)=0} . It is clear that {displaystyle f'} is a G-linear map, because the sum or difference of G-linear maps is also G-linear. Then we return to the above argument, where we used the fact that a map was G-linear to conclude that the kernel is a subrepresentation, and is thus either zero or equal to all of {displaystyle V} ; because it is not zero (it contains {displaystyle x} ) it must be all of V and so {displaystyle f'} is trivial, so {displaystyle f=lambda I} .

Corollary of Schur's Lemma An important corollary of Schur's lemma, having applications in quantum information theory, is as follows: Corollary : Using the same notation from the previous theorem, let {displaystyle h} be a linear mapping of V into W, and set {displaystyle h_{0}={frac {1}{|G|}}sum _{gin G}(rho _{W}(g))^{-1}hrho _{V}(g)} If {displaystyle V} and {displaystyle W} are not isomorphic, then {displaystyle h_{0}=0} . If {displaystyle V=W} is finite-dimensional over an algebraically closed field (e.g. {displaystyle mathbb {C} } ); and if {displaystyle rho _{V}=rho _{W}} , then {displaystyle h_{0}=I,mathrm {Tr} [h]/n} , where n is the dimension of V. That is, {displaystyle h_{0}} is a homothety of ratio {displaystyle mathrm {Tr} [h]/n} .

Proof: Let us first show that {displaystyle h_{0}} is a G-linear map, i.e., {displaystyle rho _{W}(g)circ h_{0}=h_{0}circ rho _{V}(g)} for all {displaystyle gin G} . Indeed, consider that {displaystyle (rho _{W}(g'))^{-1}h_{0}rho _{V}(g')={frac {1}{|G|}}sum _{gin G}(rho _{W}(g'))^{-1}(rho _{W}(g))^{-1}hrho _{V}(g)rho _{V}(g')={frac {1}{|G|}}sum _{gin G}(rho _{W}(gcirc g'))^{-1}hrho _{V}(gcirc g')=h_{0}} Now applying the previous theorem, for case 1, it follows that {displaystyle h_{0}=0} , and for case 2, it follows that {displaystyle h_{0}} is a scalar multiple of the identity matrix (i.e., {displaystyle h_{0}=mu I} ). To determine the scalar multiple {displaystyle mu } , consider that {displaystyle mathrm {Tr} [h_{0}]={frac {1}{|G|}}sum _{gin G}mathrm {Tr} [(rho _{V}(g))^{-1}hrho _{V}(g)]=mathrm {Tr} [h]} It then follows that {displaystyle mu =mathrm {Tr} [h]/n} .

Formulation in the language of modules If M and N are two simple modules over a ring R, then any homomorphism f: M → N of R-modules is either invertible or zero.[3] In particular, the endomorphism ring of a simple module is a division ring.[4] The condition that f is a module homomorphism means that {displaystyle f(rm)=rf(m){text{ for all }}min M{text{ and }}rin R.} 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 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.

This holds more generally for any algebra {displaystyle R} over an uncountable algebraically closed field {displaystyle k} and for any simple module {displaystyle M} that is at most countably-dimensional: the only linear transformations of {displaystyle M} that commute with all transformations coming from {displaystyle 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 {displaystyle k} -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to {displaystyle k} . This is in general stronger than being irreducible over the field {displaystyle k} , and implies the module is irreducible even over the algebraic closure of {displaystyle k} .[citation needed] 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 groups and Lie algebras. There are three parts to the result.[5] First, suppose that {displaystyle V_{1}} and {displaystyle V_{2}} are irreducible representations of a Lie group or Lie algebra over any field and that {displaystyle phi :V_{1}rightarrow V_{2}} is an intertwining map. Then {displaystyle phi } is either zero or an isomorphism.

Second, if {displaystyle V} is an irreducible representation of a Lie group or Lie algebra over an algebraically closed field and {displaystyle phi :Vrightarrow V} is an intertwining map, then {displaystyle phi } is a scalar multiple of the identity map.

Third, suppose {displaystyle V_{1}} and {displaystyle V_{2}} are irreducible representations of a Lie group or Lie algebra over an algebraically closed field and {displaystyle phi _{1},phi _{2}:V_{1}rightarrow V_{2}} are nonzero intertwining maps. Then {displaystyle phi _{1}=lambda phi _{2}} for some scalar {displaystyle 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 {displaystyle {mathfrak {g}}} is a Lie algebra and {displaystyle U({mathfrak {g}})} is the universal enveloping algebra of {displaystyle {mathfrak {g}}} . Let {displaystyle pi :{mathfrak {g}}rightarrow mathrm {End} (V)} be an irreducible representation of {displaystyle {mathfrak {g}}} over an algebraically closed field. The universal property of the universal enveloping algebra ensures that {displaystyle pi } extends to a representation of {displaystyle U({mathfrak {g}})} acting on the same vector space. It follows from the second part of Schur's lemma that if {displaystyle x} belongs to the center of {displaystyle U({mathfrak {g}})} , then {displaystyle pi (x)} must be a multiple of the identity operator. In the case when {displaystyle {mathfrak {g}}} is a complex semisimple Lie algebra, an important example of the preceding construction is the one in which {displaystyle x} is the (quadratic) Casimir element {displaystyle C} . In this case, {displaystyle pi (C)=lambda _{pi }I} , where {displaystyle lambda _{pi }} is a constant that can be computed explicitly in terms of the highest weight of {displaystyle pi } .[6] The action of the Casimir element plays an important role in the proof of complete reducibility for finite-dimensional representations of semisimple Lie algebras.[7] See also Schur complement.

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 of M.

A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For the important class of modules of finite length, the following properties are equivalent (Lam 2001, §19): 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. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring of integers, the module of rational numbers 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 A5 over the finite field with three elements F3 has F3 as its endomorphism ring.

See also Quillen's lemma Notes ^ Schur, Issai (1905). "Neue Begründung der Theorie der Gruppencharaktere" [New foundation for the theory of group characters]. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (in German). Berlin: Preußische Akademie der Wissenschaften: 406–432. ^ Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Graduate Texts in Mathematics. Vol. 42. New York, NY: Springer. p. 13. doi:10.1007/978-1-4684-9458-7. ISBN 978-1-4684-9458-7. ^ Sengupta 2012, p. 126 ^ Lam 2001, p. 33 ^ Hall 2015 Theorem 4.29 ^ Hall 2015 Proposition 10.6 ^ Hall 2015 Section 10.3 References Dummit, David S.; Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: Wiley. p. 337. ISBN 0-471-36857-1. Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings. Berlin, New York: Springer-Verlag. ISBN 978-0-387-95325-0. Sengupta, Ambar (2012). Representing finite groups: a semisimple introduction. New York. doi:10.1007/978-1-4614-1231-1_8. ISBN 9781461412311. OCLC 769756134. Shtern, A.I.; Lomonosov, V.I. (2001) [1994], "Schur lemma", Encyclopedia of Mathematics, EMS Press Categories: Representation theoryLemmas