In representation theory (group theory), an irreducible representation is a nontrivial representation with no nontrivial subrepresentations. Similarly, an irreducible module is another name for a simple module.
In the theory of manifolds, an n-manifold is irreducible if any embedded (n-1) sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.
The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over S^1 and the twisted 2-sphere bundle over S^1.
An irreduciblerepresentation of a group is a group representation that has no nontrivial invariant subspaces.
In a given representation, reducible or irreducible, the group characters of all matrices belonging to operations in the same class are identical (but differ from those in other representations).
The number of irreduciblerepresentations of a group is equal to the number of conjugacy classes in the group.
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