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The support of the reduced Kronecker coefficients satisfies the size triangle inequalities

Statement

If ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ then $∣ α ∣ \leq ∣ β ∣ + ∣ γ ∣, ∣ β ∣ \leq ∣ α ∣ + ∣ γ ∣, ∣ γ ∣ \leq ∣ α ∣ + ∣ β ∣$ (Murnaghan's inequalities). Hence the support $Supp (α, β)$ of the nonzero reduced coefficients with fixed $α, β$ lies in the triangle region of the weight lattice.

Facts & Assumptions

Given: Partitions $α, β, γ$ with ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ and a nonzero summand $(μ, λ, η, ρ, θ, π)$ of the expansion.

[F1]

${\overset{ˉ}{g}}_{α, β}^{γ} = \sum c_{μ, θ, π}^{γ} c_{ρ, λ, θ}^{α} c_{ρ, η, π}^{β} g_{λ, η}^{μ}$ , a sum of nonnegative integers, and for every nonzero summand $∣ α ∣ = ∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣$ , $∣ β ∣ = ∣ ρ ∣ + ∣ η ∣ + ∣ π ∣$ , $∣ γ ∣ = ∣ μ ∣ + ∣ θ ∣ + ∣ π ∣$ and $∣ μ ∣ = ∣ λ ∣ = ∣ η ∣$ (The nonnegative expansion of a reduced Kronecker coefficient).

Proof

technique · direct

1.1F1

Since ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ is a sum of nonnegative integers, some summand is positive; fix such a summand, whose four coefficients are positive and whose sizes satisfy the degree equations of [F1].

2.1F1step 1.1

The three differences. Using the degree equations and $∣ λ ∣ = ∣ η ∣ = ∣ μ ∣$ , compute $∣ α ∣ + ∣ β ∣ - ∣ γ ∣ = (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) + (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) - (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) = 2 ∣ ρ ∣ + ∣ λ ∣ + ∣ η ∣ - ∣ μ ∣ = 2 ∣ ρ ∣ + ∣ μ ∣ \geq 0$ ; next $∣ α ∣ + ∣ γ ∣ - ∣ β ∣ = (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) + (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) - (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) = ∣ λ ∣ + ∣ μ ∣ + 2 ∣ θ ∣ - ∣ η ∣ = ∣ μ ∣ + 2 ∣ θ ∣ \geq 0$ ; and symmetrically $∣ β ∣ + ∣ γ ∣ - ∣ α ∣ = (∣ ρ ∣ + ∣ η ∣ + ∣ π ∣) + (∣ μ ∣ + ∣ θ ∣ + ∣ π ∣) - (∣ ρ ∣ + ∣ λ ∣ + ∣ θ ∣) = ∣ μ ∣ + 2 ∣ π ∣ \geq 0$ . Each difference is a sum of nonnegative integers, giving the three inequalities for this summand.

3.1F1step 2.1∎

Since these inequalities depend only on the sizes of $α, β, γ$ and not on the chosen positive summand, they hold whenever ${\overset{ˉ}{g}}_{α, β}^{γ} > 0$ , that is, for every $γ \in Supp (α, β)$ . This is precisely the assertion that the support lies in the triangle region cut out by the three size inequalities. The argument is a finite computation with nonnegative integers and uses no choice principle; the empty partition is allowed, and then the corresponding inequality is the trivial nonnegativity bound.

Depends on

The nonnegative expansion of a reduced Kronecker coefficient

Used by

The exact global stabilization threshold for a fixed Kronecker product Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emmanuel Briand, Rosa Orellana and Mercedes Rosas, The stability of the Kronecker product of Schur functions, arXiv:0907.4652v2 (2009); Murnaghan's inequalities, p. 4 (standard reference, not scraped)