Decomposing Reducible Representations with the Great Orthogonality Theorem

Welcome to the fifth lesson in our module on Representation Theory. In the last session, we introduced the concept of the character as a basis-invariant "fingerprint" of a representation, constant across conjugacy classes. We saw how to compute characters from known matrices and assemble them into a character table, which offers a compact summary of a group's representation theory.

Today, we will unlock the true computational power of this formalism. We will introduce the great orthogonality theorem for characters, a profound result that treats the rows of the character table as orthogonal vectors in a specific vector space. You will learn to apply this theorem to achieve a central goal in the application of group theory to physics: decomposing any reducible representation into its fundamental, irreducible components. This procedure allows us to predict how symmetries dictate the structure of physical systems, such as the splitting of energy levels in a crystal, without ever needing to write down a single representation matrix.

1. The Structure of Reducible Representations

As we've discussed, a reducible representation is one for which a basis can be found that brings all representation matrices simultaneously into the same block-diagonal form. Each of these blocks is itself an irreducible representation (irrep).

A visual analogy for representation decomposition. A general, reducible representation matrix might look complicated. The goal of reduction is to find a change of basis (a similarity transformation) that reveals its underlying structure, making it block-diagonal (c). Each block corresponds to an irreducible representation. Our task is to determine the composition of these blocks.

This means that any reducible representation, let's call it $Ψ$ , can be expressed as a direct sum of irreps $Γ_{i}$ :

Ψ = n_{1} Γ_{1} \oplus n_{2} Γ_{2} \oplus \dots = i ⨁ n_{i} Γ_{i}

Here, the integer $n_{i}$ is the multiplicity, telling us how many times the irrep $Γ_{i}$ appears in the block-diagonal structure of $Ψ$ .

Since the trace of a block-diagonal matrix is the sum of the traces of its blocks, this relationship holds for the characters as well:

χ^{(Ψ)} (g) = i \sum n_{i} χ^{(Γ_{i})} (g)

Our goal is to find the coefficients $n_{i}$ . To do this, we need a tool that can "project out" a specific irreducible component from this sum. That tool is the great orthogonality theorem.

2. The Great Orthogonality Theorem for Characters

In the previous lesson, you saw hints of a hidden structure within character tables. This structure is formally captured by the great orthogonality theorem. While it derives from a more general theorem concerning individual matrix elements, the version for characters is the most useful for practical applications.

The theorem states that the characters of two non-equivalent irreducible representations are orthogonal. More precisely, for any two irreps $Γ_{i}$ and $Γ_{j}$ of a finite group $G$ :

g \in G \sum χ^{(Γ_{i})} (g)^{*} χ^{(Γ_{j})} (g) = N_{G} δ_{ij}

where $N_{G}$ is the order of the group and $g$ is summed over all elements. Since characters are constant on conjugacy classes, we can write this more efficiently as a sum over the classes $C$ :

C \sum N_{C} χ^{(Γ_{i})} (C)^{*} χ^{(Γ_{j})} (C) = N_{G} δ_{ij}

Here, $N_{C}$ is the number of elements in class $C$ . This theorem tells us that if we treat the characters of the irreps as vectors in a space of class functions, these vectors form an orthogonal set.

[PDF] Crystal-Field Theory, Tight-Binding Method and Jahn-Teller Effect

This document by Pavarini provides a concise statement of the orthogonality relations and their context.

Please read the paragraph starting with "A matrix representation Γ" to review the concepts of reducible and irreducible representations. Then, focus on the paragraph starting "However the trace", which introduces the character table and formally states the orthogonality relations in equations (12) and (13). Note that equation (12) corresponds to the "row orthogonality" we are discussing here.

3. The Reduction Formula

With the orthogonality theorem in hand, we can now derive a formula for the multiplicities $n_{i}$ . The method is a classic technique you've likely seen in the context of Fourier series or orthogonal polynomials. We start with the character decomposition:

χ^{(Ψ)} (C) = i \sum n_{i} χ^{(Γ_{i})} (C)

Now, we multiply by $N_{C} χ^{(Γ_{j})} (C)^{*}$ and sum over all classes $C$ :

C \sum N_{C} χ^{(Γ_{j})} (C)^{*} χ^{(Ψ)} (C) = C \sum N_{C} χ^{(Γ_{j})} (C)^{*} (i \sum n_{i} χ^{(Γ_{i})} (C))

We can swap the order of summation on the right-hand side:

C \sum N_{C} χ^{(Γ_{j})} (C)^{*} χ^{(Ψ)} (C) = i \sum n_{i} (C \sum N_{C} χ^{(Γ_{j})} (C)^{*} χ^{(Γ_{i})} (C))

The term in the parenthesis is precisely what the orthogonality theorem evaluates for us. It is equal to $N_{G} δ_{ij}$ . The sum over $i$ therefore collapses, leaving only the term where $i = j$ :

C \sum N_{C} χ^{(Γ_{j})} (C)^{*} χ^{(Ψ)} (C) = n_{j} N_{G}

Rearranging gives us the master equation for decomposing a reducible representation, often called the reduction formula:

n_{j} = \frac{1}{N _{G}} C \sum N_{C} χ^{(Γ_{j})} (C)^{*} χ^{(Ψ)} (C)

This remarkable formula tells us that to find how many times an irrep $Γ_{j}$ is contained in a reducible representation $Ψ$ , we simply need to compute the "dot product" of their characters and divide by the order of the group.

Contents

These notes from UC San Diego provide a rigorous derivation of this formula.

Please read the subsection "2.4.1 Reducible representations". It starts from the decomposition of the character \chi_\Psi in equation (2.59) and, using the character orthogonality relation, derives the explicit formula for the multiplicities n_\Gamma(\Psi) in equation (2.61). This is the exact formula we will be using.

4. A Worked Example: $C_{3 v}$

Let's apply this formula to the $Γ_{x y z}$ representation of the $C_{3 v}$ group (isomorphic to $S_{3}$ ) that we encountered in the last lesson. The character table is:

$C_{3 v}$	$E$	$2 C_{3}$	$3 σ_{v}$
$A_{1}$	1	1	1
$A_{2}$	1	1	-1
$E$	2	-1	0

The character of the reducible representation $Γ_{x y z}$ (formed by the action of the symmetry operations on the $(x, y, z)$ basis vectors) was found to be $χ^{(xyz)} = (3, 0, 1)$ . The order of the group is $N_{G} = 1 + 2 + 3 = 6$ . Now we apply the reduction formula to find the multiplicities $n_{A_{1}}$ , $n_{A_{2}}$ , and $n_{E}$ .

L6A Reducible Representations

The video "L6A Reducible Representations" by Emily Tsui gives an excellent, clear, step-by-step walkthrough of this exact calculation.

First, watch the segment that introduces the formula from 3:47 to 6:03. Then, watch her apply it to find the coefficients for the A_1, A_2, and E representations from 6:03 to 9:48.

Let's summarize the calculation shown in the video:

For $A_{1}$ :
$n_{A_{1}} = \frac{1}{6} [(1 \cdot 1^{*} \cdot 3) + (2 \cdot 1^{*} \cdot 0) + (3 \cdot 1^{*} \cdot 1)] = \frac{1}{6} [3 + 0 + 3] = 1$
For $A_{2}$ :
$n_{A_{2}} = \frac{1}{6} [(1 \cdot 1^{*} \cdot 3) + (2 \cdot 1^{*} \cdot 0) + (3 \cdot (- 1)^{*} \cdot 1)] = \frac{1}{6} [3 + 0 - 3] = 0$
For $E$ :
$n_{E} = \frac{1}{6} [(1 \cdot 2^{*} \cdot 3) + (2 \cdot (- 1)^{*} \cdot 0) + (3 \cdot 0^{*} \cdot 1)] = \frac{1}{6} [6 + 0 + 0] = 1$

The result is that $Γ_{x y z} = 1 \cdot A_{1} \oplus 0 \cdot A_{2} \oplus 1 \cdot E$ , or more simply, $Γ_{x y z} = A_{1} \oplus E$ . The 3D representation breaks down into a 1D irrep ( $A_{1}$ ) and a 2D irrep ( $E$ ). This confirms what one can find by explicitly block-diagonalizing the matrices, but we did it with far less work.

A Quick Test for Irreducibility

A useful consequence of the orthogonality relation is a simple test for whether a given representation is irreducible. For an irrep $Γ_{j}$ , the "norm-squared" of its character is:
$\sum_{C} N_{C} ∣ χ^{(Γ_{j})} (C) ∣^{2} = N_{G}$
For a general representation $Ψ = ⨁_{i} n_{i} Γ_{i}$ , one can show that:
$\sum_{C} N_{C} ∣ χ^{(Ψ)} (C) ∣^{2} = N_{G} \sum_{i} n_{i}^{2}$
This leads to a powerful criterion: A representation $Ψ$ is irreducible if and only if $\frac{1}{N _{G}} \sum_{C} N_{C} ∣ χ^{(Ψ)} (C) ∣^{2} = 1$ . If this value is greater than 1, the representation is reducible.

5. Application in Condensed Matter: Crystal Field Splitting

This machinery is not just a mathematical curiosity; it is a predictive tool. One of its classic applications in your field of interest, condensed matter physics, is crystal field theory.

In an isolated atom, the electronic states are eigenfunctions of angular momentum, and all $2 l + 1$ orbitals for a given $l$ (e.g., the five d-orbitals) are degenerate due to the full rotational symmetry of the SO(3) group. When this atom is placed in a crystal, it is subjected to an electric field (the "crystal field") produced by the neighboring ions. This field no longer has SO(3) symmetry but rather the smaller point group symmetry of the crystal lattice site.

The original $(2 l + 1)$ -dimensional representation, which was irreducible under SO(3), becomes a reducible representation of the smaller point group. Decomposing this representation tells us precisely how the energy levels will split and what the degeneracies of the new, split levels will be.

[PDF] Crystal-Field Theory, Tight-Binding Method and Jahn-Teller Effect

The final section of the Pavarini PDF provides a worked example of this physical application. It analyzes how atomic orbitals split in a cubic crystal field, which has the symmetry of the point group O.

Please study the section titled "3 Crystal-field theory", starting from the introductory paragraphs through to the end of the calculations for the s, p, d, and f shells. Focus on these steps in the text: The character table for the point group O is given in equation (20). The text then provides the characters for the reducible representations \Gamma^s, \Gamma^p, \Gamma^d, \Gamma^f under the symmetry operations of the group O. You don't need to derive these characters for now; just take them as given. Follow the final step where the author applies the decomposition formula to find how each shell splits. For example, for the d-orbitals (l=2), the character is \chi^{(d)} = (5, -1, 1, 1, -1) for the classes (E, 8C_3, 3C_2, 6C_2', 6C_4). Applying the reduction formula yields the decomposition \Gamma^d = E \oplus T_2.

The result $Γ^{d} = E \oplus T_{2}$ is a physical prediction. Consulting the character table for group O, we see that the irrep $E$ is 2-dimensional and $T_{2}$ is 3-dimensional. This tells us that the 5-fold degenerate d-orbitals split into a 2-fold degenerate level (called $e_{g}$ ) and a 3-fold degenerate level (called $t_{2 g}$ ) in a cubic crystal environment. The same procedure shows that p-orbitals remain degenerate ( $Γ^{p} = T_{1}$ , a 3D irrep), while f-orbitals split into three levels ( $Γ^{f} = A_{2} \oplus T_{1} \oplus T_{2}$ ).

Conclusion

In this lesson, you've mastered one of the most powerful computational techniques in applied group theory.

Key Takeaways:

The great orthogonality theorem for characters states that the characters of irreducible representations form an orthogonal basis for class functions.
This theorem gives rise to the reduction formula, $n_{j} = \frac{1}{N _{G}} \sum_{C} N_{C} χ_{j}^{*} χ_{Ψ}$ , which allows for the systematic decomposition of any reducible representation into its irreducible components.
This decomposition is not just a mathematical formality; it directly predicts physical phenomena, such as the splitting of atomic energy levels in a crystal field, a cornerstone of condensed matter physics and inorganic chemistry.

So far, we have taken the character tables of the irreps as given. But how are these tables constructed in the first place? In our next lesson, we will see how the very same orthogonality relations can be used as a set of constraints to bootstrap the construction of a group's complete character table, often from very little initial information.

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