nLab irreducible representation

Idea

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

Similarly for irreducible modules.

Definition

Given some algebraic structure, such as a group, equipped with a notion of (linear) representation, 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 representations.

Notice that 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 an indecomposable representation but still reducible.

In good cases for finite dimensional representations, the two notions (irreducible, indecomposable) coincide.

Examples

Example

(irreducible real linear representations of cyclic groups)

For $n \in \mathbb{N}$, $n \geq 2$, the isomorphism classes of irreducible real linear representations of the cyclic group $\mathbb{Z}/n$ are given by precisely the following:

1. the 1-dimensional trivial representation $\mathbf{1}$;

2. the 1-dimensional sign representation $\mathbf{1}_{sgn}$;

3. the 2-dimensional standard representations $\mathbf{2}_k$ of rotations in the Euclidean plane by angles that are integer multiples of $2 \pi k/n$, for $k \in \mathbb{N}$, $0 \lt k \lt n/2$;

   hence the restricted representations of the defining real rep of SO(2) under the subgroup inclusions $\mathbb{Z}/n \hookrightarrow SO(2)$, hence the representations generated by real $2 \times 2$ trigonometric matrices of the form

   $$\rho_{\mathbf{2}_k}(1) \;=\; \left( \array{ cos(\theta) & -sin(\theta) \\ sin(\theta) & \phantom{-}cos(\theta) } \right) \phantom{AA} \text{with} \; \theta = 2 \pi \tfrac{k}{n} \,,$$

(For $k = n/2$ the corresponding 2d representation is the direct sum of two copies of the sign representation: $\mathbf{2}_{n/2} \simeq \mathbf{1}_{sgn} \oplus \mathbf{1}_{sgn}$, and hence not irreducible. Moreover, for $k \gt n/2$ we have that $\mathbf{2}_{k}$ is irreducible, but isomorphic to $\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}$).

In summary:

$$Rep^{irr}_{\mathbb{R}} \big( \mathbb{Z}/n \big)_{/\sim} \;=\; \big\{ \mathbf{1}, \mathbf{1}_{sgn}, \mathbf{2}_k \;\vert\; 0 \lt k \lt n/2 \big\}$$

(e.g. tom Dieck 09 (1.1.6), (1.1.8))

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Related concepts

• reducible representation

• irreducible character

• simple object

• fundamental representation

• In quantum physics an irreducible representation of an algebra of observables is called a superselection sector.

  In Wigner classification the irreps of the spacetime symmetry groups are identified with elementary particle species.

References

See any text on representation theory, for instance

• Tammo tom Dieck, Representation theory, 2009 (pdf)

See also

• Wikipedia, Irreducible representation