On (almost) extreme components in Kronecker products of characters of the symmetric groups. (Q403129)

This paper considers the \textit{Kronecker problem} for the symmetric group, which asks for the decomposition of the tensor product of two irreducible characters into irreducibles. This problem appears to be extremely difficult, and continues to attract a great deal of attention. The paper concentrates in particular on analysing cases where the tensor product has few homogeneous components; the main result is a classification (verifying a conjecture of the first author and Kleshchev) of cases where the tensor product has no more than four homogeneous components. This is achieved by a careful analysis of the \textit{extreme} and \textit{almost extreme} constituents of the tensor product: a constituent \([\lambda]\) of the product \([\mu].[\nu]\) is extreme if \(\lambda_1\) equals \(|\mu\cap\nu|\) or \(\lambda'_1\) equals \(|\mu\cap\nu'|\) (these being the largest possible values for \(\lambda_1\) and \(\lambda'_1\), respectively), and is almost extreme if \(\lambda_1=|\mu\cap\nu|-1\) or \(\lambda_1'=|\mu\cap\nu'|-1\). The authors classify all tensor products \([\mu].[\nu]\) with no more than four (almost) extreme constituents, from which it is quite easy to deduce the main result.NEWLINENEWLINE The proofs of the results are surprisingly elementary; the main tool is a result due to Dvir, which enables the (recursive) calculation of the Kronecker coefficients using Littlewood-Richardson coefficients. The nature of the recursion involved makes this result particularly helpful for finding the (almost) extreme constituents of a tensor product. The authors import a few other background results on Kronecker coefficients, but the paper is for the most part self-contained, relying only on partition combinatorics.NEWLINENEWLINE The paper is very well written, breaking down a long combinatorial proof into manageable sections in an appealing way. The writing is very careful, and I had to look hard to find a typo. Although easy to follow, the paper would not necessarily be a good introductory read for this subject area, since the background material and discussion is (sensibly) kept to a minimum. But it will be interesting to see whether this paper inspires further results on the Kronecker problem for the symmetric groups.