Pseudodeterminants and Perfect Square Spanning Tree Counts

Abstract

The pseudodeterminant pdet(M) of a square matrix is the last nonzero coe cient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If @ is a symmetric or skewsymmetric matrix then pdet(@@t) = pdet(@)2. Whenever @ is the kth boundary map of a self-dual CWcomplex X, this linear-algebraic identity implies that the torsion-weighted generating function for cellular k-trees in X is a perfect square. In the case that X is an antipodally self-dual CW-sphere of odd dimension, the pseudodeterminant of its kth cellular boundary map can be interpreted directly as a torsion-weighted generating function both for k-trees and for (k ð€€€ 1)-trees, complementing the analogous result for evendimensional spheres given by the second author. The argument relies on the topological fact that any self-dual even-dimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.