Reducible vs. Irreducible Representations

In our previous lesson, we established the formal connection between abstract symmetry groups and the concrete linear operators that act on the state vectors of a quantum system. We defined a group representation as a homomorphism into a group of matrices and saw that different choices of basis lead to equivalent representations. This naturally leads to a crucial question: can we simplify a given representation? Just as a chemical compound can be broken down into its constituent elements, we want to know if a representation can be decomposed into more fundamental, "atomic" building blocks.

This lesson addresses that question by introducing the concepts of reducible and irreducible representations. You will learn to distinguish between these two types, which is the foundational step in classifying all possible ways a symmetry can manifest in a physical system. Understanding this distinction allows us to organize the vast landscape of representations into a manageable set of elementary components, the irreducible representations, or "irreps."

1. The Anatomy of a Representation: Reducibility and Direct Sums

The core idea is that some representations are composites of smaller, independent representations. Imagine a vector space that can be split into two (or more) subspaces, where the symmetry operations only ever mix vectors within a given subspace, never between them. If such a split is possible, the representation is called reducible.

In matrix terms, this means there exists a change of basis (a similarity transformation) that simultaneously brings all the representation matrices $D (g)$ into a block-diagonal form.

Spinors for Beginners 18: Irreducible Representations of SU(2) (Ladder Operators)

The video "Irreducible Representations of SU(2)" by eigenchris provides a very clear and intuitive introduction to this idea.

Watch the segment from 3:54 to 5:13. The presenter explains how one can "cheat" to create a larger representation by combining smaller ones using a direct sum, which results in a block-diagonal matrix. This process is the essence of reducibility.

As the video explains, a representation is reducible if it is equivalent to a representation where all matrices are in block-diagonal form. The act of combining smaller representations $ρ_{1}$ and $ρ_{2}$ into a larger one is called the direct sum, denoted $ρ = ρ_{1} \oplus ρ_{2}$ . The resulting matrices have the form:

D (g) = (D_{1} (g) 0 0 D_{2} (g))

Conversely, a representation that cannot be brought into a block-diagonal form by any similarity transformation is called an irreducible representation (irrep). These are the fundamental building blocks we are seeking.

Chapter 17 - Group Theory

The textbook chapter by Arfken, Weber, and Harris provides a formal definition and a powerful, explicit example.

First, read the paragraph defining reducibility. Note the crucial point that a reducible representation may not initially appear in block-diagonal form; it must be transformable into one. Next, study Example 17.2.2: A Reducible Representation (in its entirety). This is a perfect illustration. It starts with a 3x3 representation of the group D_3 where the matrices are dense. It then provides a specific unitary transformation matrix V that, when applied via U' = VUV^{-1}, converts all the representation matrices into the same block-diagonal structure (a 1x1 block and a 2x2 block).

This example makes the concept concrete. The original 3x3 representation was reducible. After applying the correct transformation, it was decomposed into the direct sum of a 1-dimensional representation and a 2-dimensional representation. The 2-dimensional part, as the text notes, is itself irreducible because the matrices within that block do not all commute and thus cannot be simultaneously diagonalized.

2. Invariant Subspaces: The Formal View

The existence of a block-diagonal form is a consequence of a deeper structural property: the existence of an invariant subspace.

A subspace $W$ of a vector space $V$ is said to be invariant (or stable) under a representation $ρ$ if every vector in $W$ is mapped to another vector within $W$ by all operators in the representation. That is, for every $w \in W$ and every $g \in G$ , the vector $ρ (g) w$ is also in $W$ .

If a representation acting on $V$ possesses a non-trivial invariant subspace $W$ (meaning $W$ is not the zero vector or the whole space $V$ ), the representation is reducible. The block-diagonal form arises when we choose a basis for $V$ that starts with a basis for $W$ . The upper-left block is the representation restricted to the subspace $W$ , and the lower-right block is the representation on the "rest" of the space.

3. Complete Reducibility: The Physicist's Guarantee

This raises a critical question. If we find an invariant subspace $W$ , we can split our representation. But what about the remaining part? Is it guaranteed to also be an invariant subspace? In other words, can every reducible representation be completely decomposed until only irreps remain?

For the groups most relevant to physics, the answer is a resounding yes. This is formalized by Maschke's Theorem, which states that for finite groups (and, importantly, for compact Lie groups like SO(3) and SU(2)), every reducible representation is completely reducible. This means it can be written as a direct sum of irreducible representations.

The proof of this theorem is highly instructive. For unitary representations, which Wigner's theorem tells us are the standard for quantum mechanics, the proof is particularly elegant. It relies on showing that if a subspace $W$ is invariant, its orthogonal complement $W^{⊥}$ is also invariant.

Symmetries and quantum mechanics: Irreducible representations, characters, Schur's lemma

The video by Tobias Osborne, "Symmetries and quantum mechanics," provides a clear, physics-oriented proof of this concept.

First, watch the alternative proof presented from 17:41 to 27:16. This section demonstrates that for a unitary representation, if a subspace W is stable, its orthogonal complement W^\perp is also stable. The argument relies on the properties of unitary matrices, making the block-diagonal structure manifest. This is the essential ingredient for complete reducibility in quantum mechanics. Next, watch the proof of the main decomposition theorem from 44:16 to 57:48. The lecturer argues that since we can always split a reducible representation into two smaller, stable parts (V = W \oplus W^\perp), we can simply repeat this process on W and W^\perp until we are left with only irreducible pieces. The argument proceeds by induction on the dimension of the representation.

Maschke's theorem is a physicist's license to focus almost exclusively on irreps. It guarantees that these atomic components are all we need to construct and understand any representation of the symmetry groups we typically encounter.

4. The Physical Significance: Degeneracy and Hamiltonians

The distinction between reducible and irreducible representations is not just a mathematical convenience; it lies at the heart of quantum mechanics, explaining the structure of energy levels and the nature of degeneracy.

Consider a quantum system with a Hamiltonian $H$ that is invariant under a symmetry group $G$ .

The set of all eigenfunctions corresponding to a single energy eigenvalue $E$ forms a vector subspace, $V_{E}$ .
Because $H$ commutes with all the symmetry operations $ρ (g)$ in $G$ , this subspace $V_{E}$ is invariant under the action of the group. If $ψ$ is an eigenfunction with energy $E$ , then $ρ (g) ψ$ must also be an eigenfunction with the same energy $E$ .
Therefore, the set of degenerate eigenfunctions for a given energy level forms a basis for a representation of the symmetry group $G$ .

Now, what if this representation were reducible? It would mean that the space of degenerate eigenfunctions $V_{E}$ could be split into two smaller, non-interacting invariant subspaces. The symmetry operations would never mix states from one subspace into the other. This implies there is no symmetry reason for these two sets of states to have the same energy. If they do, the degeneracy is considered accidental.

Barring such accidental degeneracies, we arrive at a profound conclusion: The set of eigenfunctions corresponding to a degenerate energy level forms a basis for an irreducible representation of the Hamiltonian's symmetry group. The dimension of the irrep is equal to the degree of the degeneracy.

Chapter 17 - Group Theory

The same textbook chapter provides an excellent discussion of this physical connection.

Read the section titled "Symmetry and Physics". This section formally develops the argument that eigenfunctions of a symmetric Hamiltonian can be classified into sets that form bases for irreps. Then, read Example 17.3.1: An Even Hamiltonian. This simple example perfectly clarifies the idea. A Hamiltonian is even, H(x)=H(-x). Its symmetry group is C_s = \{I, \sigma\}, where \sigma is reflection. This group has two 1D irreps: a symmetric one (A_1) and an antisymmetric one (A_2). The eigenfunctions of H must be either even (transforming as A_1) or odd (transforming as A_2). A general function that is neither even nor odd forms a basis for a reducible representation, which can be decomposed into its even and odd parts—precisely the bases for the irreps.

Conclusion

In this lesson, we have dissected the concept of a group representation, separating them into two fundamental types. This distinction is the bedrock upon which the practical application of group theory in physics is built.

Here are the essential takeaways:

A reducible representation is a composite object. It acts on a vector space that contains a non-trivial invariant subspace, and it can be transformed into a block-diagonal form via a similarity transformation.
An irreducible representation (irrep) is an "atomic" object. It has no non-trivial invariant subspaces and cannot be simplified further.
For the groups most common in physics (finite and compact Lie groups), any representation is completely reducible, meaning it can be expressed as a direct sum of irreps.
The physical manifestation of an irrep is a set of degenerate quantum states. The dimension of the irrep corresponds to the degree of degeneracy enforced by the system's symmetry.

We now understand that irreps are the fundamental entities we need to study. But how do we characterize them? How can we tell different irreps apart? And how can we methodically decompose a reducible representation into its irrep components? These questions lead us to the powerful mathematical tools of representation theory. In our next lesson, we will begin exploring these tools by studying Schur's lemmas, two deceptively simple yet incredibly powerful theorems that govern the structure of irreps.

Recommended resources

1. The Anatomy of a Representation: Reducibility and Direct Sums

2. Invariant Subspaces: The Formal View

3. Complete Reducibility: The Physicist's Guarantee

4. The Physical Significance: Degeneracy and Hamiltonians

Conclusion