Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

Details Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

2010-10-19

arxiv.org/abs/1010.3860v1

Mathematics

Combinatorics

Scientific paper

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.

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