Request PDF on ResearchGate | Le lemme de Schur pour les représentations orthogonales | Let σ be an orthogonal representation of a group G on a real. Statement no. Condition, Conclusion in abstract formulation for vector spaces: \ rho_1: G \to GL(V_1), \rho_2: G \ are linear representations of G. Ensuite nous démontrons un lemme (le théorème II) qui est fondamental pour pour la convexité S en généralisant et précisant quelques résultats de Schur.

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Even for group rings, there are examples when the characteristic of the field divides the order of the zchur In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras. Retrieved from ” https: A representation of G with no subrepresentations other than itself and zero is an irreducible representation.

They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M. There are three parts to the result.

Archive ouverte HAL – Le lemme de Schur pour les représentations orthogonales.

Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. Irreducible representations, like the prime numbers, or like elmme simple groups in group theory, are the building blocks of representation theory. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Views Read Edit View history.

In general, Schur’s lemma cannot be reversed: Such a homomorphism is called a representation of G on V. This is in general stronger than being irreducible over the field kand implies the module is irreducible even over the algebraic closure of k.

From Wikipedia, the sxhur encyclopedia.

The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups. Thus the endomorphism ring of the module M is “as small as possible”. Schur’s lemma is frequently applied in the following particular case. In other words, we require that f commutes with the action of G. Suppose f is a nonzero G -linear map from V to W.


If M is finite-dimensional, this division algebra is finite-dimensional. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest.

Schur’s lemma – Wikidata

Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i. However, even over the ring of integersthe module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers.

A module is said to be strongly indecomposable if its endomorphism ring is a local ring. We say W is stable under Gor stable under the action of G. For other uses, see Schur’s lemma disambiguation.

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We will prove that V and W are isomorphic. G -linear maps are the morphisms in the category of representations of G. A representation on V is a special case of a group action on Vbut rather than permit any arbitrary permutations of the underlying set of Vwe restrict ourselves to invertible linear transformations.

By assumption it is not zero, so it is surjective, in which lekme it is an isomorphism. We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras.

Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier. This holds more lemmw for any algebra R over an uncountable algebraically closed field k and for any simple module M that is at most countably-dimensional: When W has this property, we call W with the given representation a subrepresentation of V.


A simple module over k -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. If M and N are two simple modules over a ring Rthen any homomorphism f: It is easy to check that this is a subspace.

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 a theorem that describes what G -linear maps can exist between two irreducible representations of G. As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional. The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G.

Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group.

Le lemme de Schur pour les représentations orthogonales.

By using this site, you agree to the Terms of Use and Privacy Policy. In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.

This page was last edited on 17 Augustat