Abstract: A minimal matrix is a matrix $A$ with non-negative integer entries such that the sequence of its entries, arranged in non-increasing order, is minimal (in the dominance order of partitions) among all sequences obtained from matrices that have the same row and column sum vectors as $A$. Minimal matrices first appeared in a characterization of 3-dimensional (0,1)-matrices that are uniquely determined by its plane sums.
It is a major open problem to give a satisfactory description of the multiplicity of a complex irreducible character of the symmetric group $S_n$ in the Kronecker product of other two irreducible characters of $S_n$. In this talk we explain how to use minimal matrices to produce minimal components in Kronecker products of irreducible characters. A theorem due to E. Snapper plays an important role in some results presented in this talk.
This talk will be accessible to graduate students.