Views Read Edit View history. 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 representation of G with no subrepresentations other than itself and zero is an irreducible representation. We will prove that V and W are isomorphic. 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.
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Duktilar A module is said to be strongly indecomposable if its endomorphism ring is a local ring. In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.
Irreducible representations, like the prime numbers, or like the simple groups in group theory, are the building blocks of representation theory. When W has this property, we call W with the given representation a subrepresentation of V. G -linear maps are the morphisms in the category of representations of G. From Wikipedia, the free encyclopedia.
There are three parts to the result. Such a homomorphism is called a representation of G on V. It is easy to check that this is a subspace. 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. 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.
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. This page was last edited on 17 Augustat If k is the field of complex numbers, the only option is that this division algebra is the complex numbers.
Suppose f is a nonzero G -linear map from V to W. However, even over the ring of integerslemms module of rational numbers has an endomorphism ring that schu a division ring, specifically the field of rational numbers. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. Even for group rings, there are examples when the characteristic of the field divides the order of the group: By assumption it is not zero, so it is surjective, in which case it is an isomorphism.
If M is finite-dimensional, this division algebra is finite-dimensional. We will prove that V and W are isomorphic. 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: As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional. Many of the initial questions and theorems of representation theory deal with the properties of irreducible dw.
They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M. Lemme de Schur 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 representation of G with no subrepresentations other than itself and zero is an irreducible representation. TOP Related Articles.
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