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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It is easy to check that this is a subspace. When W has this property, we call W with the given representation a subrepresentation of V.

Schur’s lemma – Wikipedia

The sfhur module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. In other words, we require that f commutes with the action of G. Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier.

Suppose f is a nonzero G -linear map from V to W. In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras. By using this site, you agree to the Terms of Use and Privacy Policy. 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.

Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. We say W is stable under Gor stable under the action of G.

If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory.


This lemmd more generally for any algebra R over an uncountable algebraically closed field k and for any simple module M that is at most countably-dimensional: When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. For other uses, see Schur’s lemma disambiguation. This is in general stronger than being irreducible over the field scchurand implies the module is irreducible even over the algebraic closure of k.

The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics sfhur the representation theory of finite groups.

In general, Schur’s lemma cannot be reversed: Such a homomorphism is called a representation of Schue on V. 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.

Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k. Thus the endomorphism ring of the module M is “as lemje as possible”.

Le lemme de Schur pour les représentations orthogonales.

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 If M is finite-dimensional, this division algebra is finite-dimensional. As a simple corollary of the second statement is that every complex irreducible lemmme of an Abelian group is one-dimensional. We will prove that V and W are isomorphic.

There are three parts to the 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.

We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras. Representation lem,e is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i. A representation of G with no subrepresentations other than itself and zero is an irreducible representation.


Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G. From Wikipedia, the free encyclopedia. Schur’s lemma is frequently applied in the following particular case.

A simple module over k -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

Schur’s lemma

By assumption it is not zero, so it is surjective, in which case it is an isomorphism. 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.

Retrieved from ” https: Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. Even for lemje rings, there are examples when the characteristic of the field divides the order of the group: G -linear maps are the morphisms in the category of representations of G.

If M and N are two simple modules over a ring Rthen any homomorphism f: Views Read Edit View history. A module is said to be strongly indecomposable if its endomorphism ring is a local ring.