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

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 the three partitions Furthermore, if the first parts of partitions lambda, mu are big enough then the coefficients of the Kronecker product vertical bar lambda vertical bar vertical bar mu vertical bar = Sigma(nu)g(lambda, mu, nu)vertical bar nu vertical bar do not depend on the first part but only on the other parts. The reduced Kronecker product vertical bar lambda vertical bar(center dot) * vertical bar mu vertical bar(center dot) can be viewed (roughly) as the Kronecker product vertical bar(n - vertical bar lambda vertical bar . lambda)vertical bar vertical bar(n - vertical bar mu vertical bar . mu)vertical bar 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. Furthermore, we 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 where the factors of the products are the same. (C) 2010 Elsevier Ltd. All rights reserved.