Viewing Determinants as Nonintersecting Lattice Paths yields Classical Determinantal Identities Bijectively

Markus Fulmek

Abstract


In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot-method and the Jacobi Trudi identity together with elementary observations.

After some preparations, this point of view provides "graphical proofs'' for classical determinantal identities like the Cauchy-Binet formula, Dodgson's condensation formula, the Plücker relations, Laplace's expansion and Turnbull's identity. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.


Keywords


determinantal identities; nonintersecting lattice paths; Schur function identities

Full Text: PDF