Tensor representation

Let $Γ$ be the defining representation of $G L 𝑁 (ℂ)$ . A tensor representation of $G L 𝑁 (ℂ)$ is a subrepresentation of $Γ \otimes 𝑛$ for some $𝑛 \in ℕ$ . lie Weyl’s construction associates every tensor irrep for fixed $𝑁$ to a Young diagram.

Weyl’s construction

Let $𝑉 = ℂ 𝑁$ , $G L 𝑁 (ℂ)$ act on $𝑉 \otimes 𝑛$ by $Γ \otimes 𝑛$ and $𝑆 𝑛$ act on $𝑉 \otimes 𝑛$ by the permutation representation $𝐷$ . In addition let $ℂ [𝑆 𝑛]$ act by the corresponding ∗-representation.

Let $𝑌$ be the set of Young diagrams of $𝑛$ boxes and at most $𝑁$ rows. For each $𝜆 \in 𝑌$ let $𝐿 𝜆 = ℂ [𝑆 𝑛] * 𝔢 𝜆$ be the corresponding minimal left ideal with basis ${𝑓 𝛽 𝜆} 𝑑 𝜆 𝛽 = 1$ . For a given $| 𝑣 ⟩ \in 𝑉 \otimes 𝑛$ , ${𝑓 𝛽 𝑣} 𝑑 𝜆 𝛽 = 1$ either vanish or transform in the irrep $𝐷 𝜆$ (see below). Let ${| 𝑣 𝛼 𝜆 ⟩} 𝑚 𝜆 𝛼 = 1$ be a complete¹ set of tensors such that each $𝑓 𝛽 𝜆 | 𝑣 𝛼 𝜆 ⟩$ is unique. Then

| 𝜆, 𝛼, 𝛽 ⟩ = 𝑓 𝛽 𝜆 | 𝑣 𝛼 𝜆 ⟩

form an irreducible orthonormal basis under both $G L 𝑁 (ℂ)$ and $𝑆 𝑛$ , where

𝑇 𝜆 (𝛼) = s p a n {𝜆, 𝛼, 𝛽} 𝑑 𝜆 𝛽 = 1

is a $𝑑 𝜆$ -dimensional irreducible invariant subspace under $𝑆 𝑛$ and

𝑇' 𝜆 (𝛽) = s p a n {𝜆, 𝛼, 𝛽} 𝑚 𝜆 𝛼 = 1

is a $𝑚 𝜆$ -dimensional irreducible invariant subspace transforming under $G L 𝑁 (ℂ)$ in an irrep henceforth labeled $Γ 𝜆$ . Thus

𝑉 \otimes 𝑛 = ⨁ 𝜆 \in 𝑌 𝑚 𝜆 ⨁ 𝛼 = 1 𝑇 𝜆 (𝛼) = ⨁ 𝜆 \in 𝑌 𝑑 𝜆 ⨁ 𝛽 = 1 𝑇' 𝜆 (𝛽)

where $𝑑 𝜆$ is given by the Hook length formula and $𝑚 𝜆$ is given by Stanley’s hook content formula.

Proof of vanishing property

For $| 𝑣 ⟩ \in 𝑇 𝜆 (𝛼)$ there exists some $𝑟 \in 𝐿 𝜆$ s.t. $| 𝑣 ⟩ = 𝑟 | 𝑣 𝛼 𝜆 ⟩$ . Then $𝑝 | 𝑣 𝛼 𝜆 ⟩ = 𝑝 𝑟 | 𝑣 𝛼 𝜆 ⟩ \in 𝑇 𝜆 (𝛼)$ for $𝑝 \in 𝑆 𝑛$ since $𝑝 𝑟 \in 𝐿 𝜆$ , giving invariance. Clearly ${𝑓 𝛽 𝜆 | 𝑣 𝛼 𝜆 ⟩} 𝑑 𝜆 𝛽 = 1$ is a basis of $𝑇 𝜆 (𝛼)$ . We define a matrix representation $𝐷 𝜆 𝑖 𝑗$ by
$𝑝 𝑓 𝑖 𝜆 = 𝑓 𝑗 𝜆 𝐷 𝜆 𝑗 𝑖 (𝑝)$
with summation convention. Then
$𝑝 𝑓 𝑖 𝜆 | 𝑣 𝛼 𝜆 ⟩ = 𝑓 𝑗 𝛼 𝐷 𝜆 𝑗 𝑖 (𝑝) | 𝑣 𝛼 𝜆 ⟩ = 𝑓 𝑗 𝜆 | 𝑣 𝛼 𝜆 ⟩ 𝐷 𝜆 𝑗 𝑖 (𝑝)$
as required.

Proof

proof See 07 Tensor method for constructing irreps of GL(n) and subgroups for discussion.