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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Such a homomorphism is called a representation of G on V. In other words, we require that f commutes with the action of G.

Schur’s lemma – Wikipedia

We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras. Thus the endomorphism ring of the module M is “as small as possible”. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M. It is easy to check that this is a subspace.

Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. In general, Schur’s lemma cannot be reversed: This page was last edited on 17 Augustat Retrieved from ” https: As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G.


Le lemme de Schur pour les représentations orthogonales.

By using this site, you agree to the Terms of Use and Privacy Policy. 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.

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

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.

G -linear maps are the morphisms in the category of representations of G.

In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras. There are three parts to the result.

A representation of G with no subrepresentations other schhr itself and zero is an irreducible representation.

A simple module over k -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k.

Schur’s lemma

Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G. By assumption it is not zero, so it is surjective, in which case it is an isomorphism.

Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory. In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. From Wikipedia, the free encyclopedia. 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: Suppose f is a nonzero G -linear map from V to W.


If M is finite-dimensional, this division algebra is finite-dimensional. Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i.

When W has this property, we call W with the given representation a subrepresentation of V. If M and N are two simple modules over a ring Rthen any homomorphism f: We will prove that V and W are isomorphic.

Schur’s lemma is frequently applied in the following particular case. This is in general stronger than being irreducible over the field kand 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 of the representation theory of finite groups. Even for group rings, there are examples when the characteristic of the field divides the order of the group: Views Read Edit View history.