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 module is said to be strongly indecomposable if its endomorphism ring is a local ring. Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i. Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory. In general, Schur’s lemma cannot be reversed: As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional.

If M and N are two simple modules over a ring Rthen any homomorphism f: Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k. By assumption it is not zero, so it is surjective, in which case it is an isomorphism. For other uses, see Schur’s lemma disambiguation. 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 M is finite-dimensional, this division algebra is finite-dimensional.

Schur’s lemma

This page was last edited on 17 Augustat In mathematicsSchur’s lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.

The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple.

Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. We now describe Schur’s lemma as it is usually stated in the context of representations of Lie groups and Lie algebras. This is in general stronger than being irreducible over the field kand implies the module is irreducible even over the algebraic closure of k.

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

Suppose f is a nonzero G -linear map from V to W. In other words, we require that f commutes with the action of G.

A simple module over k -algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. Views Read Edit View history. Schur’s lemma is frequently applied in the following particular case. 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. 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. Retrieved from ” https: A representation of G with no subrepresentations other than itself and zero is an irreducible representation.

In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. G -linear maps are the morphisms in the category of representations of G.

It is easy to check that this is a subspace. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still lemme particular interest.

Lemme de Schur, , X ,

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. Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier.

From Wikipedia, the free encyclopedia.