Column expansion identities and quadratic spanning forest identities

TL;DR

This paper explores column expansion identities of determinants to derive quadratic spanning forest identities, providing a combinatorial interpretation and settling a conjecture, with implications for quantum field theory calculations.

Contribution

It establishes the dimension of the space of quadratic spanning forest identities and offers a new combinatorial interpretation, resolving a conjecture from 2012.

Findings

Determined the dimension of quadratic spanning forest identity space.

Provided a combinatorial edge-swapping interpretation.

Connected identities to quantum field theory applications.

Abstract

Column expansion identities of determinants give a source of quadratic spanning forest polynomial identities and allow us determine the dimension of the space of certain quadratic spanning forest identities, settling a conjecture of one of us with Vlasev from 2012. Furthermore, we give a combinatorial interpretation of such spanning forest identities via an edge-swapping argument previously developed by one of us in 2019. Quadratic spanning forest polynomials identities are of particular interest because they are useful for quantum field theory calculations in four dimensions.