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Difference between revisions of "Irreducible representation"

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A (linear) representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526301.png" /> of a group (algebra, ring, semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526302.png" /> in a vector space (or topological vector space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526303.png" /> with only two (closed) invariant subspaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526305.png" />. Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526306.png" /> is a representation in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526307.png" /> and if it is irreducible as a representation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526308.png" /> considered as a vector space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052630/i0526309.png" /> is called an algebraically-irreducible representation. An algebraically-irreducible representation is also topologically irreducible; the converse is not true, in general. There are a number of concepts close to that of an irreducible representation, among them an [[Operator-irreducible representation|operator-irreducible representation]], and a completely-irreducible representation (one for which the family of operators forms a completely-irreducible set, cf. also [[Completely-reducible set|Completely-reducible set]]). A completely-irreducible representation is (topologically) irreducible and operator irreducible; the converse assertions are not true, in general.
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A (linear) representation $\pi$ of a group (algebra, ring, semi-group) $X$ in a vector space (or topological vector space) $E$ with only two (closed) invariant subspaces, $(0)$ and $E$. Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation. If $\pi$ is a representation in a topological vector space $E$ and if it is irreducible as a representation in $E$ considered as a vector space, then $\pi$ is called an algebraically-irreducible representation. An algebraically-irreducible representation is also topologically irreducible; the converse is not true, in general. There are a number of concepts close to that of an irreducible representation, among them an [[Operator-irreducible representation|operator-irreducible representation]], and a completely-irreducible representation (one for which the family of operators forms a completely-irreducible set, cf. also [[Completely-reducible set|Completely-reducible set]]). A completely-irreducible representation is (topologically) irreducible and operator irreducible; the converse assertions are not true, in general.

Latest revision as of 10:13, 13 April 2014

A (linear) representation $\pi$ of a group (algebra, ring, semi-group) $X$ in a vector space (or topological vector space) $E$ with only two (closed) invariant subspaces, $(0)$ and $E$. Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation. If $\pi$ is a representation in a topological vector space $E$ and if it is irreducible as a representation in $E$ considered as a vector space, then $\pi$ is called an algebraically-irreducible representation. An algebraically-irreducible representation is also topologically irreducible; the converse is not true, in general. There are a number of concepts close to that of an irreducible representation, among them an operator-irreducible representation, and a completely-irreducible representation (one for which the family of operators forms a completely-irreducible set, cf. also Completely-reducible set). A completely-irreducible representation is (topologically) irreducible and operator irreducible; the converse assertions are not true, in general.

How to Cite This Entry:
Irreducible representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_representation&oldid=31667
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article