How to Determine if a Given Group Representation is Reducible or Irreducible

Determining whether a given group representation is reducible or irreducible is a fundamental task in representation theory. This process involves a series of steps and tools that help mathematicians and researchers understand the structure of group actions on vector spaces. In this article, we will explore the necessary definitions and procedures to make this determination.

Definitions

To begin, let's define some key terms:

Group Representation

A group representation is a homomorphism from a group ( G ) to the general linear group ( GL(V) ) of a vector space ( V ). Essentially, it assigns to each element of the group a linear transformation on the vector space.

Reducible Representation

A representation is reducible if there exists a nontrivial invariant subspace ( W subset V ) such that ( V W oplus W' ) for some subspace ( W' ). In simpler terms, this means there is a subset of ( V ) that remains unchanged under the action of every element in ( G ).

Irreducible Representation

A representation is irreducible if the only invariant subspaces are the trivial subspace ( {0} ) and the whole space ( V ).

Procedure to Determine Reducibility

The process of determining whether a given representation is reducible or irreducible involves several steps:

1. Identify the Representation

Let ( rho: G to GL(V) ) be your representation, where ( V ) is a finite-dimensional vector space over a field, often the complex numbers ( mathbb{C} ) or the real numbers ( mathbb{R} ).

2. Find Invariant Subspaces

To check for reducibility, you need to look for subspaces ( W subset V ) such that ( rho(g)(W) subset W ) for all ( g in G ). This means that the action of every group element leaves the subspace ( W ) invariant.

3. Check for Nontrivial Invariant Subspaces

If you can find a nontrivial invariant subspace ( W ), i.e., ( W eq {0} ) and ( W eq V ), then the representation is reducible. If the only invariant subspaces are the trivial subspace ( {0} ) and the whole space ( V ), then the representation is irreducible.

4. Use Characters if Applicable

If the representation is finite-dimensional over ( mathbb{C} ), you can compute the character of the representation. If the character is constant on conjugacy classes, the representation is likely reducible. More formally, if the representation is reducible, the character can often be expressed as a sum of characters of irreducible representations.

5. Decompose the Representation if Needed

If you suspect the representation is reducible, you can attempt to decompose it into irreducible components. This can be done using methods such as:

Schur's Lemma

If ( W ) is an invariant subspace and ( T: W to W ) is a linear transformation that commutes with all ( rho(g) ), then ( T ) is a scalar multiple of the identity if ( W ) is irreducible.

Direct Sum Decomposition

Explicitly find a basis for ( V ) such that the action of ( G ) on ( V ) can be expressed as a direct sum of actions on the invariant subspaces.

Example

Consider a representation of a group ( G ) on ( mathbb{R}^2 ). If you can find a line through the origin (a 1-dimensional subspace) that is invariant under the action of all group elements, then the representation is reducible. If no such line exists, then it is irreducible.

Conclusion

In summary, to check if a representation is reducible, look for nontrivial invariant subspaces. If you find them, the representation is reducible. If not, it is irreducible. Additional tools like characters and Schur's Lemma can also aid in the analysis, particularly in more complex cases.