On the Combinatorics of Pfaffian Ideals

Abstract

The main topic is to the study of Pfaffian ideals in combinatorics and commutative algebra. The Pfaffian of a skew-symmetric matrix is a classical and ubiquitous object in mathematics. In this work, we want to investigate the ideal generated by the Pfaffians of skew-symmetric submatrices of a skew-symmetric matrix of variables. To be more precise, we want to investigate the combinatorics of initial ideals of that ideal. We will be interested in square-free initial ideals, which can then be considered Stanley-Reisner ideals of simplicial complexes. In the work of Jonsson and Welker, it has been shown that there is one such simplicial complex that has a description through sets of diagonals of n-gons for which no t intersect pairwise. In our work, we will derive this result and add details to the original proof. We will then also study how the initial ideal is affected if some of the entries of the skew symmetric matrix are set to zero.