Schur's lemma

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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.

Representation theory of groups

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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  , 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): VV is also an automorphism of W, ρ(g): WW, 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.

Just as we are interested in homomorphisms between groups, and in continuous maps between topological spaces, 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 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,  . 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

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Theorem (Schur's Lemma): Let V and W be vector spaces; and let   and   be irreducible representations of G on V and W respectively.[2]

  1. If   and   are not isomorphic, then there are no nontrivial G-linear maps between them.
  2. If   finite-dimensional over an algebraically closed field (e.g.  ); and if  , 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   is a nonzero G-linear map from   to  . We will prove that   and   are isomorphic. Let   be the kernel, 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 G-linear, 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 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 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  ; because it is not zero (it contains  ) it must be all of V and so   is trivial, so  .

Corollary of Schur's Lemma

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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,

  1. If   and   are not isomorphic, then  .
  2. 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.[3] In the context of molecular orbital theory, it is used to restrict atomic orbital interactions based on the molecular symmetry.[4]

Formulation in the language of modules

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Theorem: If M and N are two simple modules over a ring R, then any homomorphism f: MN of R-modules is either invertible or zero.[5] In particular, the endomorphism ring of a simple module is a division ring.[6]

The condition that f is a module homomorphism means that

 

Proof: It suffices to show that   is either zero or surjective and injective. We first show that both   and   are  -modules. If   we have  , hence  . Similarly, if  , then   for all  . Now, since   and   are submodules of simple modules, they are either trivial or equal  , respectively. If  , its kernel cannot equal   and must therefore be trivial (hence   is injective), and its image cannot be trivial and must therefore equal   (hence   is surjective). Then   is bijective, and hence an isomorphism. Consequently, every homomorphism   is either zero or invertible, which makes   into a division ring.

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.

More generally, if   is an algebra over an algebraically closed field   and   is a simple  -module satisfying   (the cardinality of  ), then  .[7] 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 to  . This is in general stronger than being irreducible over the field  , and implies the module is irreducible even over the algebraic closure of  . [citation needed]

Application to central characters

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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   is the group algebra of a finite group  , the same conclusion follows. Here, the center of   consists of elements of the shape   where   is a class function, i.e. invariant under conjugation. Since the set of class functions is spanned by the characters   of the irreducible representations  , the central character is determined by what it maps   to (for all  ). Since all   are idempotent, they are each mapped either to 0 or to 1, and since   for two different irreducible representations, only one   can be mapped to 1: the one corresponding to the module  .

Representations of Lie groups and Lie algebras

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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.[8]

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 of the second statement is that every complex irreducible representation of an abelian group is one-dimensional.

Application to the Casimir element

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Suppose   is a Lie algebra and   is the universal enveloping algebra of  . Let   be an irreducible representation of   over an algebraically closed field. The universal property 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, 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  .[9] The action of the Casimir element plays an important role in the proof of complete reducibility for finite-dimensional representations of semisimple Lie algebras.[10]

Generalization to non-simple modules

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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

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Notes

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  1. ^ 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.
  2. ^ 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.
  3. ^ Scott, A J (2006-10-27). "Tight informationally complete quantum measurements". Journal of Physics A: Mathematical and General. 39 (43): 13507–13530. arXiv:quant-ph/0604049. Bibcode:2006JPhA...3913507S. doi:10.1088/0305-4470/39/43/009. hdl:10072/22680. ISSN 0305-4470. S2CID 33144766.
  4. ^ Bishop, David M. (January 14, 1993). Symmetry and Chemistry. Dover Publications. ISBN 978-0486673554.
  5. ^ Sengupta 2012, p. 126
  6. ^ Lam 2001, p. 33
  7. ^ Bourbaki, Nicolas (2012). "Algèbre: Chapitre 8". Éléments de mathématique (Revised and expanded ed.). Springer. p. 43. ISBN 978-3031192920.
  8. ^ Hall 2015 Theorem 4.29
  9. ^ Hall 2015 Proposition 10.6
  10. ^ Hall 2015 Section 10.3

References

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