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The Racah--Speiser tensor-product algorithm

Statement

Assume the Axiom of Choice. Let $λ, μ \in Λ^{+}$ . For every weight $φ$ of $L (μ)$ with multiplicity $m_{μ} (φ) > 0$ put $ψ = φ + λ$ and say that $φ$ is regular relative to $λ$ when $ψ + ρ$ is fixed by no reflection of $W$ , equivalently $⟨ ψ + ρ, α^{\lor} ⟩ \neq 0$ for every root $α$ . If $φ$ is regular relative to $λ$ , there is a unique $u \in W$ with $u (ψ + ρ)$ strictly dominant (Finite Weyl closed chambers and stabilizers), and then $ν (φ) : = u (ψ + ρ) - ρ = u \cdot ψ$ is a dominant integral weight (the shifted action being $u \cdot ξ = u (ξ + ρ) - ρ$ ). The Racah--Speiser algorithm computes the multiplicities of Steinberg's tensor-product multiplicity formula as $c_{λ μ}^{ν} = \sum_{\begin{array}{c} φ weight of L (μ) \\ φ regular relative to λ, ν (φ) = ν \end{array}} (- 1)^{ℓ (u (φ))} m_{μ} (φ),$ the sum being finite; a weight $φ$ for which $ψ + ρ$ is not regular is discarded, and every $ν \in Λ^{+}$ with $c_{λ μ}^{ν} \neq 0$ occurs as $ν (φ)$ for some regular weight $φ$ of $L (μ)$ .

Facts & Assumptions

Given: AC, dominant integral weights $λ, μ, ν$ , the weight set of $L (μ)$ with multiplicities $m_{μ} (φ) = \dim L (μ)_{φ}$ (Weight and weight space), and the Weyl group $W$ acting on $h^{*}$ by the reflection action (Root reflections and the Weyl group action).

[F1]

Steinberg's formula: $c_{λ μ}^{ν} = \sum_{w \in W} (- 1)^{ℓ (w)} m_{μ} (w^{- 1} (ν + ρ) - (λ + ρ)),$ the sum being finite (Steinberg's tensor-product multiplicity formula).

[F2]

The shifted dominant weight $ν + ρ$ is strictly dominant, and $⟨ ρ, α_{i}^{\lor} ⟩ = 1$ (Positive coroot pairings of a dominant integral weight). Every orbit in the real root span $E$ has exactly one point in the closed dominant chamber. Its stabilizer is generated by reflections in the simple walls through that point. Consequently a regular $ξ \in E$ has a strictly dominant representative and a unique element $u$ sending it there; uniqueness of the element is asserted only for regular points (Finite Weyl closed chambers and stabilizers, Integral, dominant, and strictly dominant weights, Finite Weyl root system, lattice and chamber conventions). In particular $ξ = φ + λ + ρ$ is integral; if regular, all simple-coroot pairings of $u ξ$ are positive integers, so subtracting $ρ$ leaves nonnegative integral pairings and $u ξ - ρ \in Λ^{+}$ .

[F3]

Length parity: $ℓ (w s) \equiv ℓ (w) + 1 (m o d 2)$ for every reflection $s \in W$ (the sign $ε (w) = (- 1)^{ℓ (w)}$ is a homomorphism), so $(- 1)^{ℓ (w s)} = - (- 1)^{ℓ (w)}$ (The sign of the Weyl length is multiplicative).

Proof

1.1F1givenalgebra

Rewrite the Steinberg sum by the second argument. For $w \in W$ put $φ_{w} : = w^{- 1} (ν + ρ) - (λ + ρ)$ and $ψ_{w} + ρ : = w^{- 1} (ν + ρ) = φ_{w} + λ + ρ$ , so that $c_{λ μ}^{ν} = \sum_{w} (- 1)^{ℓ (w)} m_{μ} (φ_{w})$ by [F1], the sum being finite. Fix a weight $φ$ of $L (μ)$ and let $W (φ) = {w \in W : φ_{w} = φ} = {w : w^{- 1} (ν + ρ) = ψ + ρ}$ where $ψ = φ + λ$ . This set is nonempty exactly when $ψ + ρ$ is $W$ -conjugate to $ν + ρ$ .

1.2F1F2givenalgebra

Regular weights contribute one term each. Suppose $φ$ is regular, so that $ψ + ρ$ has trivial stabilizer. If $W (φ) \neq \emptyset$ , then $ν + ρ = w (ψ + ρ)$ is strictly dominant for any $w \in W (φ)$ , so by the uniqueness in [F2] there is exactly one such $w$ , namely $w = u (φ)$ where $u (φ)$ is the unique element with $u (φ) (ψ + ρ)$ strictly dominant; in that case $ν (φ) = u (φ) (ψ + ρ) - ρ = ν$ , and the contribution of $φ$ to $c_{λ μ}^{ν}$ is $(- 1)^{ℓ (u (φ))} m_{μ} (φ)$ . If $W (φ) = \emptyset$ , or if $ν (φ) \neq ν$ , the weight $φ$ contributes nothing to $c_{λ μ}^{ν}$ .

2.1F1F3step 1.1algebra

Irregular weights cancel. Suppose $φ$ is not regular: $ψ + ρ$ is fixed by some reflection $s$ . Then $W (φ)$ is stable under right multiplication by $s$ , because $(w s)^{- 1} (ν + ρ) = s w^{- 1} (ν + ρ) = s (ψ + ρ) = ψ + ρ$ for every $w \in W (φ)$ ; the map $w \mapsto w s$ is a fixed-point-free involution of $W (φ)$ , and by [F3] the signs of paired terms are opposite. Since the multiplicity $m_{μ} (φ)$ is the same for paired terms, the total contribution of the group $W (φ)$ to the sum of step 1.1 is $0$ .

3.1F1step 1.1step 1.2step 2.1algebra∎

Combining step 1.2 and 2.1, the value of $c_{λ μ}^{ν}$ is the sum of $(- 1)^{ℓ (u (φ))} m_{μ} (φ)$ over the regular weights $φ$ of $L (μ)$ with $ν (φ) = ν$ , which is the displayed Racah--Speiser formula; the sum is finite because $L (μ)$ has finitely many weights. If $c_{λ μ}^{ν} \neq 0$ , the displayed sum is nonzero, so at least one regular weight $φ$ satisfies $ν (φ) = ν$ ; this proves the final assertion.

Depends on

Used by

The Clebsch--Gordan tensor decomposition for sl2 Example

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Sources

A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser 2002 (standard reference, not scraped)
D. E. Littlewood and A. R. Richardson / G. Racah and J. Speiser, as presented in R. Goodman and N. R. Wallach, Symmetry, Representations, and Invariants, GTM 255, Springer 2009 (standard reference, not scraped)
P. Etingof, Lie Groups and Lie Algebras II (MIT 18.755, Spring 2024), complete lecture notes (standard reference, not scraped)