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Schur's lemma is a fundamental result in representation theory, an elementary observation about irreducible modules, which is nonetheless noteworthy because of its profound applications.
Proof. Note that both the kernel,  , and the image,
 , are  -submodules of  and  , respectively. Since  is irreducible,  is either trivial or all of  . In the former case,
 is all of  -- also because  is irreducible -- and hence  is invertible. In the latter case,  is the zero map. 
One of the most important consequences of Schur's lemma is the following.
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"Schur's lemma" is owned by rmilson. [ full author list (2) | owner history (1) ]
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Cross-references: words, eigenvalue, linear transformation, ground field, multiplication, scalar, field, algebraically closed, finite-dimensional, consequences, image, kernel, zero map, invertible, homomorphism, finite group, applications, irreducible modules, representation
There are 3 references to this entry.
This is version 19 of Schur's lemma, born on 2002-11-04, modified 2006-06-12.
Object id is 3570, canonical name is SchursLemma.
Accessed 7899 times total.
Classification:
| AMS MSC: | 20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous) | | | 20C15 (Group theory and generalizations :: Representation theory of groups :: Ordinary representations and characters) |
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Pending Errata and Addenda
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