Determinants and Perfect Matchings

TL;DR

This paper provides a combinatorial interpretation of the determinant using Brauer diagrams, linking permutation signs to crossings and explaining properties of antisymmetric matrices and Pfaffians.

Contribution

It introduces a novel combinatorial perspective on determinants via Brauer diagrams, connecting permutation signs to diagram crossings.

Findings

Determinant interpreted as a generating function over Brauer diagrams

Sign of permutation replaced by crossings in diagrams

Explains why even antisymmetric matrices' determinants are squares of Pfaffians

Abstract

We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.