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 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.
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. In other words, the only linear transformations of M that lemm with all transformations coming from R are scalar multiples of the identity.
When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G. By assumption it is not zero, so it is scchur, in which case it is an isomorphism.
Lemme de Schur
If M and N are two simple modules over a ring Rthen any homomorphism f: It is easy to check that this is a subspace. 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. In general, Schur’s lemma cannot be reversed: G -linear maps are the morphisms in the category of 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.
This is in general stronger than being irreducible over the field kand implies the module is irreducible even over the algebraic closure of k. This page was last edited on 17 Augustat Irreducible representations, like sschur prime numbers, or like the simple groups in group theory, are the building blocks of representation theory. Suppose f is a nonzero G -linear map from V to W.
There are three parts to the result. Thus the endomorphism ring of the module M is “as small as possible”. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. From Wikipedia, the free encyclopedia.
We will prove that V and W are isomorphic. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers.
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We say W is stable under Gor stable under the action of G. Views Read Edit View history. A module is said to be strongly indecomposable if its endomorphism ring is a local ring.
Even for group rings, there are examples when the characteristic of the field divides the order of the group: This holds more generally for any algebra R lekme an uncountable algebraically closed field k and for any simple module M that re at most countably-dimensional: Then Schur’s lemma says that the endomorphism ring of the module M is a division algebra over the field k.
Archive ouverte HAL – Le lemme de Schur pour les représentations orthogonales.
Schur’s lemma is frequently applied in the following particular case. Many of the initial questions and theorems of representation theory deal with the properties of irreducible representations. Retrieved from ” https: Representation theory is the study of homomorphisms from a group, Ginto the general linear group GL V of a vector space V ; i.
Such a homomorphism is called a representation of G on V. Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G. When W has this property, we call W with lsmme given representation a subrepresentation of V. A representation of G with no subrepresentations other than itself and zero is an irreducible representation.
If M is finite-dimensional, this division algebra is finite-dimensional. In mathematicsSchur’s lemma  is an elementary but extremely useful statement in representation theory of groups and algebras.
As 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.
The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics lemmme the representation theory of finite groups. The one module version of Schur’s lemma admits generalizations involving modules M that are not necessarily simple.
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.