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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From Achur, the free encyclopedia. The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. Schur’s lemma is frequently applied in the following particular case.

Schur’s lemma – Wikidata

Suppose f is a nonzero G -linear map from V to Scuur. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i.

Even for group rings, there are examples when the characteristic of the field divides the order of the group: If M is finite-dimensional, this division algebra is finite-dimensional.

We say W is stable under Gor stable under the action of G. As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G. There are three parts to the result. Views Read Edit View history. df


In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. When W has this property, we call W with the given representation a subrepresentation of V. In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.

Lemme de Schur

Such a homomorphism is called a representation of G on V. 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. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings schhr as the complex group ring of a finite group.

A representation of G with no subrepresentations other than itself and zero is an irreducible representation. We will prove that V and W are isomorphic. By using this site, you agree to the Terms of Use and Privacy Policy. In other words, we require that f commutes with the action of G. Retrieved from ” https: We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras.

As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional. This page was last edited on 17 Augustat By assumption it is sschur zero, so it is surjective, in which case it is an isomorphism. In general, Schur’s lemma cannot be reversed: For other uses, see Schur’s lemma disambiguation.

Lemme de Schur, , X ,

Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier. When the field is not algebraically closed, the case where the endomorphism ring is lemke small as possible is still of particular interest. This is in general stronger than being irreducible over lemmd field kand implies the module is irreducible even over the algebraic closure of k. Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G.


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: G -linear maps are the morphisms in the category of representations of G. Thus the endomorphism ring of the module M is “as small as possible”. Le,me k is the field of complex numbers, the only option is that this division algebra is the complex numbers.

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.

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.