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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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The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group.

There are three parts to lemem result. As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G.

This holds more generally for any algebra R over an uncountable algebraically closed field k and schue any simple module M that is at most countably-dimensional: Suppose f is a nonzero G -linear map from V to W. We will prove that V and W are isomorphic.

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

Le lemme de Schur pour les représentations orthogonales.

This page was last edited on 17 Augustat Thus the endomorphism ring of the module M is “as small as possible”. 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.


G -linear maps are the morphisms in the category of representations of G. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Leme other uses, see Schur’s lemma disambiguation. Such a homomorphism is called a representation of G on V.

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

In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras. If M is finite-dimensional, this division algebra is finite-dimensional.

Schur’s lemma is frequently applied in the following particular case. It is easy to ,emme that this is a subspace. If M and N are two simple modules over a ring Rthen any homomorphism f: When W has this property, we call W with the given representation a subrepresentation of V. Retrieved from ” https: They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

Views Read Edit View history. Even for group rings, there are examples when echur characteristic of the field divides the order of the group: A representation of G with no subrepresentations other than itself and zero is an irreducible representation. In other words, we scgur that f commutes with the action of G.

Irreducible representations, like the prime numbers, or like the simple groups lfmme group theory, are the building blocks of representation theory. From Wikipedia, the free encyclopedia. However, even ve the ring of integersthe module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers.

A simple module over k -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. In general, Schur’s lemma cannot be reversed: A module is said to be strongly indecomposable if its endomorphism ring is a local ring. By assumption it is not zero, so it is surjective, in which case it is an isomorphism.


Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is schuf of particular interest.

As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional. Representation shcur is the study of homomorphisms from a group, Ginto the general linear group GL V of a schuur space V ; i.

Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier. Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k. 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.

This is in general stronger than being irreducible over the field kand implies the module is irreducible even over the algebraic closure of k. We now describe Schur’s se as it is usually stated in the context of representations of Lie groups and Lie algebras.

We say W is stable under Gor stable under the action of G. 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 dr G.