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{{otheruses|Schur's lemma (disambiguation)}}
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In [[mathematics]], '''Schur's lemma'''<ref>Issai Schur (1905) [http://books.google.com/books?id=KwUoAAAAYAAJ&pg=PA406&#v=onepage&q&f=false "Neue Begründung der Theorie der Gruppencharaktere"] (New foundation for the theory of group characters), ''Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin'', pages 406-432.</ref> is an elementary but extremely useful statement in [[representation theory]] of [[group (mathematics)|groups]] and [[algebra]]s. In the group case it says that if ''M'' and ''N'' are two finite-dimensional [[irreducible representation]]s 
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of a group ''G'' and ''&phi;'' is a linear map from ''M'' to ''N'' that commutes with the action of the group, then either ''&phi;'' is [[invertible]], or ''&phi;'' = 0. An important special case occurs when ''M''&nbsp;=&nbsp;''N'' and ''&phi;'' is a self-map. 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]]. Schur's lemma admits generalisations to [[Lie group]]s and [[Lie algebra]]s, the most common of which is due to [[Jacques Dixmier]].
 
== Formulation in the language of modules ==
 
If ''M'' and ''N'' are two [[simple module]]s over a ring ''R'', then any [[homomorphism]] ''f'': ''M'' &rarr; ''N'' of ''R''-modules is either invertible or zero. In particular, the [[endomorphism ring]] of a simple module is a [[division ring]].<ref>Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=33|text=Schur's Lemma|p. 33}}.</ref>
 
The condition that ''f'' is a module homomorphism means that
 
: <math> f(rm) = rf(m)\text{ for all }m \in M\text{ and }r \in R. \, </math>
 
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''.
 
Schur's lemma is frequently applied in the following particular case. Suppose that ''R'' is an [[algebra]] over the field '''C''' of [[complex numbers]] and ''M'' = ''N'' is a finite-dimensional simple module over ''R''. Then Schur's lemma says that the endomorphism ring of the module ''M'' is a division ring; this division ring contains '''C''' in its center, is finite-dimensional over '''C''' and is therefore equal to '''C'''.  Thus the [[endomorphism ring]] of the module ''M'' is "as small as possible". More generally, this result holds for algebras over any [[algebraically closed]] field and for simple modules that are at most countably-dimensional.  When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is of particular interest: A simple module over ''k''-algebra is said to be [[absolutely irreducible|absolutely simple]] if its endomorphism ring is isomorphic to ''k''.  This is in general stronger than being irreducible over the field ''k'', and implies the module is irreducible even over the algebraic closure of ''k''.
 
== Matrix form ==
 
Let ''G'' be a complex [[matrix group]]. This means that ''G'' is a set of square matrices of a given order ''n'' with complex entries and ''G'' is closed under [[matrix multiplication]] and inversion. Further, suppose that ''G'' is ''irreducible'': there is no [[Linear subspace|subspace]] ''V'' other than 0 and the whole space which is invariant under the action of ''G''. In other words,
 
: <math>\text{if }gV\subseteq V\text{ for all }g\text{ in }G,\text{ then either }V=0\text{ or }V=\mathbb{C}^n.</math>  
 
Schur's lemma, in the special case of a single representation, says the following. If ''A'' is a complex matrix of order ''n'' that [[Commuting matrices|commute]]s with all matrices from ''G'' then ''A'' is a [[scalar matrix]]. If ''G'' is not irreducible, then this is not true. For example, if one takes the subgroup ''D'' of diagonal matrices inside of GL(''n'','''C'''), then the center of ''D'' is ''D'', which contains non scalar matrices. As a simple corollary, every complex  irreducible representation of [[Abelian group]]s is one-dimensional.
 
See also [[Schur complement]].
 
== Generalization to non-simple modules ==
 
The one module version of Schur's lemma admits generalizations involving modules ''M'' that are not necessarily simple. They express relations between the module-theoretic properties of ''M'' and the properties of the [[endomorphism ring]] of ''M''.
 
A module is said to be '''strongly indecomposable''' if its endomorphism ring is a [[local ring]].  For the important class of modules of [[Length of a module|finite length]], the following properties are equivalent {{harv|Lam|2001|loc=&sect;19}}:
* A module ''M'' is [[indecomposable module|indecomposable]];
* ''M'' is strongly indecomposable;
* Every endomorphism of ''M'' is either nilpotent or invertible.  
 
In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a [[division ring]]. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring of [[integer]]s, the module of [[rational number]]s has an endomorphism ring that is a division ring, specifically the field of rational numbers.  Even for group rings, there are examples when the characteristic of the field divides the order of the group: the [[Jacobson radical]] of the [[projective cover]] of the one-dimensional representation of the [[alternating group]] on five points over the field with three elements has the field with three elements as its endomorphism ring. <!-- In general, [[multiplicity-free]] [[uniserial module]]s, provide a source of such modules. -->
 
== See also ==
*[[Quillen's lemma]]
 
==Notes==
{{reflist}}
 
==References==
*David S. Dummit, Richard M. Foote. ''Abstract Algebra.'' 2nd ed., pg. 337.
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-95325-0 | year=2001 }}
 
[[Category:Representation theory]]
[[Category:Lemmas]]

Latest revision as of 05:44, 22 June 2014

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