# Contents

## Idea

An irreducible representation – often abbreviated irrep – is a representation that has no smaller non-trivial representations “sitting inside it”.

## Definition

An irreducible representation is a representation that has no nontrivial proper subobject in the category of all representations in question and yet which is not itself trivial either. In other words, an irrep is a simple object in the category of representation.

## Remarks

There is also the closely related but in general different notion of an indecomposable representation?. Every irrep is indecomposable, but the converse may fail.

A representation that has proper nontrivial subrepresentations but can not be decomposed into a direct sum of such representations is indecomposable but still reducible.

Revised on May 15, 2013 14:10:29 by Urs Schreiber (82.169.65.155)