Using Grassmann calculus in combinatorics: Lindstr\"om-Gessel-Viennot lemma and Schur functions

TL;DR

This paper leverages Grassmann calculus to provide new proofs of key combinatorial identities and introduces an extended class of Schur polynomials with a convolution property.

Contribution

It introduces Grassmann calculus techniques to combinatorics and defines a one-parameter extension of Schur polynomials with a natural convolution identity.

Findings

New proofs of Lindström-Gessel-Viennot and Jacobi-Trudi identities

Definition of a one-parameter extension of Schur polynomials

Establishment of a convolution identity for the extended Schur polynomials

Abstract

Grassmann (or anti-commuting) variables are extensively used in theoretical physics. In this paper we use Grassmann variable calculus to give new proofs of celebrated combinatorial identities such as the Lindstr\"om-Gessel-Viennot formula for graphs with cycles and the Jacobi-Trudi identity. Moreover, we define a one parameter extension of Schur polynomials that obey a natural convolution identity.