Multivariable Vandermonde determinants, amalgams of matrices and Specht modules

TL;DR

This paper derives new formulas for multivariable Vandermonde determinants using Specht modules, enabling elementary proofs of properties like transfinite diameter multiplicativity.

Contribution

It introduces novel formulas for amalgamated matrices' determinants, connecting algebraic combinatorics with potential theory.

Findings

Formulas express multivariable Vandermonde determinants as sums of factorized terms.

Provides an elementary proof of the multiplicativity of transfinite diameter.

Links Specht modules to determinant calculations in matrix amalgamation.

Abstract

Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are 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.