Representation operator

Given a unitary representation $𝑈 : 𝐺 \to G L (𝑉)$ and some representation (usually an irrep) $Γ : 𝐺 \to G L (𝑉 0)$ , a representation operator¹ transforming in $Γ$ is a linear map $O : 𝑉 0 \to 𝖵 𝖾 𝖼 𝗍 ℂ (𝑉, 𝑉)$ satisfying² #m/def/rep

O (Γ (𝑔) 𝑣) = 𝑈 (𝑔) O (𝑣) 𝑈 (𝑔) †

for all $𝑔 \in 𝐺$ and $𝑣 \in 𝑉$ . If $Γ$ is an irrep $Γ 𝜇$ we denote a corresponding representation operator as $O 𝜇$ .

warning

Every operator transforms in a representation

Fixed basis

The properties and motivation for a representation operator become clearer when a basis is ${𝑒 𝑗}$ fixed for $𝑉 0$ . It is common to think of a representation operator $O$ as a set of irreducible operators³ $O 𝑗 = O (𝑒 𝑗) : 𝑉 \to 𝑉$ corresponding to each basis vector. The condition above thence becomes

𝑈 (𝑔) O 𝑖 𝑈 (𝑔) † = \sum 𝑗 O 𝑗 Γ 𝑗 𝑖 (𝑔)

which is essentially the statement that the $O 𝑖$ transform like the basis vectors $𝑒 𝑗$ under $𝐺$ . This is a direct generalisation of the Tensor operator (including scalar and vector operators), which transform in the standard representation of $S O (3)$ .

Properties

If $Γ 𝜇$ is the trivial representation, then $O 𝜇$ is just an operator that commutes with $𝑈$ .
Irreducible operators applied to an irreducible orthonormal basis transform in the product representation
Wigner-Eckart theorem

#state/develop | #lang/en | #SemBr

Keppeler refers to this as a set of irreducible operators ↩
2015, Introduction to tensors and group theory for physicists, §6.2, p. 276ff ↩
2023, Groups and representations, §4.2, p. 54 ↩

Representation operator

Fixed basis

Properties

Footnotes