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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523301.png" /> of a group (algebra, ring, semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523302.png" /> in a vector space (or topological vector space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523303.png" />''
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A vector (respectively, a closed vector) subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523304.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523305.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523306.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523307.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523308.png" /> is a projection operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i0523309.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233011.png" /> is an invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233012.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233014.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233016.png" /> is invariant for any representation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052330/i05233017.png" />; it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also [[Contraction of a representation|Contraction of a representation]]; [[Completely-reducible set|Completely-reducible set]]; [[Irreducible representation|Irreducible representation]].
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'' $  \pi $
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of a group (algebra, ring, semi-group)  $  X $
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in a vector space (or topological vector space)  $  E $''
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A vector (respectively, a closed vector) subspace $  F \subset  E $
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such that for any $  \xi \in F $
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and any $  x \in X $
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one has $  \pi ( x ) \xi \in F $.  
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If $  P $
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is a projection operator from $  E $
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onto $  F $,  
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then $  F $
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is an invariant subspace of $  \pi $
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if and only if $  P \pi ( x ) P = \pi ( x ) P $
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for all $  x \in X $.  
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The subspace $  \{ 0 \} $
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in $  E $
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is invariant for any representation in $  E $;  
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it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also [[Contraction of a representation|Contraction of a representation]]; [[Completely-reducible set|Completely-reducible set]]; [[Irreducible representation|Irreducible representation]].

Latest revision as of 22:13, 5 June 2020


$ \pi $ of a group (algebra, ring, semi-group) $ X $ in a vector space (or topological vector space) $ E $

A vector (respectively, a closed vector) subspace $ F \subset E $ such that for any $ \xi \in F $ and any $ x \in X $ one has $ \pi ( x ) \xi \in F $. If $ P $ is a projection operator from $ E $ onto $ F $, then $ F $ is an invariant subspace of $ \pi $ if and only if $ P \pi ( x ) P = \pi ( x ) P $ for all $ x \in X $. The subspace $ \{ 0 \} $ in $ E $ is invariant for any representation in $ E $; it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also Contraction of a representation; Completely-reducible set; Irreducible representation.

How to Cite This Entry:
Invariant subspace of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=47419
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article