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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Schur’s lemma – Groupprops
If M and N are two simple modules schuf a ring Rthen any homomorphism f: 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 V schjr, we restrict ourselves to invertible linear transformations. We will prove that V and W are isomorphic. A representation of G with no subrepresentations other than itself and zero is an irreducible representation.
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. 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.
If M is finite-dimensional, this division algebra is finite-dimensional. Even for group rings, there are examples when the characteristic of the field divides the order of the group: In mathematicsSchur’s lemma  is an elementary but extremely useful statement in representation theory of groups and algebras.
Schur’s lemma – Wikipedia
In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.
We say W is stable under Gor stable under the action of G. 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. Schur’s lemma dde frequently applied in the following particular case.
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Suppose f is a nonzero G -linear map from V to W. The group version is a special case of the module version, since any representation of a group G can equivalently be viewed as lemmw module over the group ring of G.
Schur’s lemma admits generalisations to Lie groups and Lie algebrasthe most common of which is due to Jacques Dixmier.
Thus the endomorphism ring of the module M is “as small as possible”. 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 module is said to scuhr strongly indecomposable if its endomorphism ring is a schut ring. Views Read Edit View history. It is easy to check that this is a subspace. The one module version of Schur’s lemma admits generalizations involving modules M that are ed necessarily simple.
Schur’s Lemma is a theorem that describes what G -linear maps can exist between two irreducible representations of G.
This page was last edited on 17 Augustat 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 zero, so it is surjective, in which case it is an isomorphism. 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, we require that f commutes with the action of G.
For other uses, see Schur’s lemma disambiguation.
G -linear maps are the morphisms in the category of representations of G. As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional. As we are interested in homomorphisms between groups, or continuous maps between topological spaces, we are interested in certain functions between representations of G.
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.