Abstract
We resolve three interrelated problems on reduced Kronecker coefficients g ¯(α,β,γ). First, we disprove the saturation property which states that g ¯(Nα,Nβ,Nγ)>0 implies g ¯(α,β,γ)>0 for all N>1. Second, we esimate the maximal g ¯(α,β,γ), over all |α|+|β|+|γ|=n. Finally, we show that computing g ¯(λ,μ,ν) is strongly #P-hard, i.e. #P-hard when the input (λ,μ,ν) is in unary.
Highlights
The reduced Kronecker coefficients were introduced by Murnaghan in 1938 as the stable limit of Kronecker coefficients, when a long first row is added: g (α, β, γ) := lim g α[n], β[n], γ[n], where α[n] := (n − |α|, α1, α2, . . .), n ≥ |α| + α1, (1)
They occupy the middle ground between the Kronecker and the LR–coefficients
While the latter are well understood and have a number of combinatorial interpretations, the former are notorious for their difficulty
Summary
The reduced Kronecker coefficients were introduced by Murnaghan in 1938 as the stable limit of Kronecker coefficients, when a long first row is added:. G (α, β, γ) := lim g α[n], β[n], γ[n] , where α[n] := They generalize the classical Littlewood–Richardson (LR–) coefficients: g As such, they occupy the middle ground between the Kronecker and the LR–coefficients. They occupy the middle ground between the Kronecker and the LR–coefficients While the latter are well understood and have a number of combinatorial interpretations, the former are notorious for their difficulty. It is generally believed that the reduced Kronecker coefficients are simpler and more accessible than the (usual) Kronecker coefficients, cf [9, 18]. ISSN (electronic) : 1778-3569 https://comptes- rendus.academie- sciences.fr/mathematique/
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have