Multivariable Vandermonde determinants, amalgams of matrices and Specht modules

Abstract

Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of a matrix which is obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.