On Vandermonde determinants via $n$-determinants

TL;DR

This paper explores the properties of Vandermonde determinants using the concept of n-determinants, connecting them to Schur polynomials and providing a new formula related to the Jacobi-Trudi identity.

Contribution

It introduces a novel approach to analyze Vandermonde determinants through n-determinants and establishes a new representation of Schur polynomials as minors of a specific matrix.

Findings

Schur polynomials can be expressed as minors of a matrix with elementary symmetric polynomial entries

The method applies to particular cases and generalizes to a broad class of determinants

A new proof of the second Jacobi-Trudi identity is provided

Abstract

We use earlier defined notion of $n$- determinant to investigate sub-determinants of an extended Vandermonde matrix. Firstly, we demonstrate our method on a number of particular cases. Then we prove that all these results may be stated in terms of Schur's polynomials. In our main result, we prove that Schur polynomials are equal to minors of a fixed matrix, which entries are formed of elementary symmetric polynomials. Such a formula is known as the second Jaccobi-Trudi identity.