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

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Summary

Introduction

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/

Saturation property
Maximal values
Complexity
Disproof of the saturation property
Bounds and complexity via identities
Final remarks and open problems
Full Text
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