Schur Polynomials through Lindstr\"om Gessel Viennot Lemma

Abstract

In this article, we use Lindstr\"om Gessel Viennot Lemma to give a short, combinatorial, visualizable proof of the identity of Schur polynomials -- the sum of monomials of Young tableaux equals to the quotient of determinants. As a by-product, we have a proof of Vandermonde determinant without words. We also prove the cauchy identity. In the remarks, we discuss factorial Schur polynomials, dual Cauchy identity and the relation beteen Newton interpolation formula.