Symmetry groups of pfaffians of symmetric matrices

TL;DR

This paper characterizes the symmetry group of the pfaffian polynomial of symmetric matrices as a dihedral group and computes specific pfaffians for matrices with particular symmetric components.

Contribution

It establishes the dihedral group symmetry of the pfaffian of symmetric matrices and explicitly computes pfaffians for matrices with squared differences and cosine-based entries.

Findings

Symmetry group of the pfaffian is dihedral.

Explicit pfaffian calculations for specific symmetric matrices.

Enhanced understanding of pfaffian symmetries in symmetric matrices.

Abstract

We prove that symmetry group of the pfaffian polynomial of a symmetric matrix is a dihedral group. We calculate pfaffians of symmetric matrices with components $(x_i-x_j)^2$ and $\cos(x_i-x_j)$ for $i<j.$