- Orthogonality Relations in Group Representation Theory

Introduction #

This document explores the orthogonality relations in group representation theory, which are fundamental in understanding how irreducible representations of finite groups behave.

Diadic Linear Operator and Transformation Properties #

We begin by defining a diadic linear operator:

When this operator is transformed by a representation matrix $D_a(g_1)$:

Let’s substitute $g’=g_1g^{-1}$, which implies $g=g’^{-1}g_1$. This gives us:

Application of Schur’s Lemma #

By Schur’s lemma, we can deduce that:

where $I_{n_a}$ is the identity matrix of dimension $n_a$.

Computing the Coefficient $\lambda^a_{jl}$ #

To compute $\lambda^a_{jl}$, we use the cyclic property of the trace: $\text{tr}(AB)=\text{tr}(BA)$.

From the original definition:

Hence:

where $N$ is the order of the group $G$.

From the Schur’s lemma form:

Comparing these two expressions: $$\lambda^a_{jl} = \frac{N}{n_a} \delta_{jl}$$

Orthogonality Relations #

Therefore: $$\sum_{g \in G} D_a(g^{-1})|a,j\rangle\langle b,l|D_b(g) = \frac{N}{n_a}\delta_{ab}\delta_{jl}I_{n_a}$$

Taking matrix elements and Considering irreducible representation unitary: $$\langle a,k|\left(\sum_{g \in G} D_a^{\dagger}(g)|a,j\rangle\langle b,l|D_b(g)\right)|b,m\rangle$$ $$ = \frac{N}{n_a}\delta_{ab}\delta_{jl}\delta_{km}$$

This leads to the orthogonality relation:

From the perspective of state vectors or functions of $g$, $\sqrt{\frac{n_a}{N}}[D_a(g)]_{jk}$ forms an orthonormal basis.

Corollary #

By counting the number of elements in the complete orthonormal basis (state vectors or functions of $g$), we obtain:

$$N = \sum_i n_i^2$$

where $N$ is the order of $G$, and $n_i$ is the dimension of the irreducible representation $i$.