Reduced Kronecker products which are multiplicity free or contain only few components

TL;DR

This paper classifies reduced Kronecker products that are multiplicity free or have few components, providing bounds on the number of constituents and analyzing their structural properties.

Contribution

It introduces a classification of reduced Kronecker products with limited components and establishes bounds on their constituents and differences.

Findings

Classified reduced Kronecker products with fewer than 10 components.

Provided lower bounds for the number of constituents and component pairs.

Showed that equality of two reduced Kronecker products implies identical factors.

Abstract

It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions \lambda,\mu are big enough then the coefficients of the Kronecker product [\lambda][\mu]=\sum_\n g(\l,\m,\n)[\nu] do not depend on the first part but only on the other parts. The reduced Kronecker product [\lambda]_\bullet \star[\mu]_\bullet can be viewed (roughly) as the Kronecker product [(n-|\lambda|,\lambda)][(n-|\mu|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free and those which contain less than 10 components.We furthermore give general lower bounds for the number of constituents and components of a given reduced Kronecker product. We also give a lower bound for the number of pairs of components whose corresponding partitions differ by one box. Finally we argue that equality of two reduced Kronecker products is only possible in the trivial case that the factors of the product are the same.