# Operator-irreducible representation

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.