Multiple hooks in Kronecker products of \(S_{n}\) characters and a conjecture of Berele and Imbo
If \(\chi^{\lambda}\) and \(\chi^{\mu}\) are complex irreducible characters of the symmetric group \(S_n\) corresponding to the partitions \(\lambda\) and \(\mu\) of \(n\), respectively, then the Kronecker product \(\chi^{\lambda}\otimes\chi^{\mu}\) is also a character of \(S_n\). It is a difficult open problem to give a satisfactory way of computing the multiplicities \(c(\lambda,\mu,\nu)\) of \(\chi^{\nu}\) in \(\chi^{\lambda}\otimes\chi^{\mu}\). A conjecture of \textit{A. Berele} and \textit{T. D. Imbo} [J. Algebra 246, 356-366 (2001; Zbl 1024.20010)] gives some information on the global behaviour of the coefficients \(c(\lambda,\mu,\nu)\) in the following way. Let the ``hook'' \(\text{ H}(k,l)\) be the set of all partitions \(\lambda=(\lambda_1,\lambda_2,\ldots)\) such that \(\lambda_j\leq l\) for all \(j>k\). The conjecture states that if \(\lambda,\mu\vdash n\) belong to \(\text{ H}(k,k)\), then the Kronecker product \(\chi^{\lambda}\otimes\chi^{\mu}\) contains at least one component \(\chi^{\nu}\) with \(\nu\in\text{ H}(k,k)\). In the paper under review the author uses results of \textit{M. Clausen} and \textit{H. Meier} [Bayreuther Math. Schr. 45, 1-17 (1993; Zbl 0799.05004)] and \textit{Y. Dvir} [J. Algebra 154, 125-140 (1993; Zbl 0848.20006)] and proves the conjecture of Berele and Imbo. The proof is constructive and the required hooks are computed combinatorially. As a by-product the author also gives constructive proofs of two theorems in the paper by Berele and Imbo.
scientific article; zbMATH DE number 1891324
scientific article