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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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 ldmme transformations. A module is said to be strongly indecomposable if its endomorphism ring is a local ring.

## Le lemme de Schur pour les représentations orthogonales.

Schur’s lemma is frequently applied in the following particular case. Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory. 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. Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k.

Suppose f is cshur nonzero G -linear map from V to W.

## Lemme de Schur

Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. 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. We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras.

Such a homomorphism is called a representation of G on V.

From Wikipedia, the free encyclopedia. There are three parts to the result. By assumption it is not zero, so it is surjective, in which case it is an isomorphism.

This holds 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: A representation of G with no subrepresentations lemmd than itself and zero is an irreducible representation. Scnur a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional.

For other uses, see Schur’s lemma disambiguation. Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier.

In general, Schur’s lemma cannot be reversed: As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G.

They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

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

Thus the endomorphism ring of the module M lemmd “as small as possible”. Even for group rings, there are examples when the characteristic of the field divides the order of the group: Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G.

If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. We say W sdhur stable under Gor stable under the action of G.

### Schur’s lemma – Wikipedia

The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. G -linear maps are the morphisms in the category of representations of G.

This page was last edited on 17 Augustat It is easy to check that this is a subspace. When the field shur not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest.

In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.