The Canada Day Theorem

TL;DR

The paper presents a new proof of the Canada Day Theorem, relating sums of minors of symmetric matrices to group actions on graph matchings, originating from integrable PDE research.

Contribution

It introduces an alternative proof of the Canada Day Theorem using group orbits on bipartite graph matchings, linking matrix minors to combinatorial group actions.

Findings

New proof elucidates the theorem's underlying mechanism

Connects matrix minors with combinatorial group actions

Provides insights from integrable PDE context

Abstract

The Canada Day Theorem is an identity involving sums of $k \times k$ minors of an arbitrary $n \times n$ symmetric matrix. It was discovered as a by-product of the work on so-called peakon solutions of an integrable nonlinear partial differential equation proposed by V. Novikov. Here we present another proof of this theorem, which explains the underlying mechanism in terms of the orbits of a certain abelian group action on the set of all $k$-edge matchings of the complete bipartite graph $K_{n,n}$.