Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

TL;DR

This paper presents a bijective, graphical approach to classical determinantal identities using nonintersecting lattice paths, simplifying proofs and introducing a potentially new generalization of Dodgson's condensation formula.

Contribution

It offers a novel bijective, graphical interpretation of determinants as nonintersecting lattice paths, providing simple proofs for classical identities and a new generalization of Dodgson's condensation.

Findings

Graphical proofs for classical determinantal identities

A new determinantal identity generalizing Dodgson's condensation

Simplification of proofs using lattice path interpretation

Abstract

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindstr\"om-Gessel-Viennot interpretation of semistandard Young tableaux and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides very simple "graphical proofs" for classical determinantal identities like the Cauchy--Binet formula, Dodgson's condensation formula, the Pl\"ucker relations and Laplace's expansion. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.