Operator-irreducible representation

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A representation $\pi$ of a group (algebra, ring, semi-group) $X$ on a (topological) vector space $E$ such that any (continuous) linear operator on $E$ commuting with every operator $\pi(x)$, $x\in X$, is a scalar multiple of the identity operator on $E$. If $\pi$ is a completely-irreducible representation (in particular, if $\pi$ is a finite-dimensional irreducible representation), then $\pi$ is an operator-irreducible representation; the converse is not always true. If $\pi$ is a unitary representation of a group or a symmetric representation of a symmetric algebra, then $\pi$ is an operator-irreducible representation if and only if $\pi$ is an irreducible representation.



[a1] I.M. Gel'fand, M.I. Graev, N.Ya. Vilenkin, "Generalized functions" , 5. Integral geometry and representation theory , Acad. Press (1966) pp. 149 ff (Translated from Russian)
[a2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) pp. 114 (Translated from Russian)
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Operator-irreducible representation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article