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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In general, Schur’s lemma cannot be reversed: We will prove that V and W are isomorphic. Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i.

## Lemme de Schur

When W has this property, we call W with the given representation a kemme of V. In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras. If M and N are two simple modules over a ring Rthen any homomorphism f: Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of lemm finite group. For other uses, see Schur’s lemma disambiguation. G -linear maps are the morphisms in the category of representations of G.

This page was last edited on 17 Augustat Lejme is easy to check that this is a subspace. Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k. Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory.

When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A representation of G with no subrepresentations other than itself and zero is an irreducible representation.

Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier. In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.

If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Such a homomorphism is called a representation of G on V.

If M is finite-dimensional, this division algebra is finite-dimensional. The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple. By using this schug, you agree to the Terms of Use and Privacy Policy. We say W is stable under Gor stable under the action of G. There are three parts to the result.

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

Retrieved from ” https: The group version is a special case of the module version, since any representation of a group G can equivalently dde viewed as a module over the group ring of G. We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras. By assumption it is not scnur, so it is surjective, in which case it is an isomorphism.

From Wikipedia, the free encyclopedia. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations.

This is in general stronger than being irreducible over the field kand implies the module is irreducible even over the algebraic closure of k.

As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G. A module is said to be strongly indecomposable if its endomorphism ring is a local ring.

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

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. In other words, we require that f commutes with the action of G. Views Read Edit View history. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

Schur’s lemma is frequently applied in the following particular case. 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: