Pseudodeterminants and perfect square spanning tree counts
Jeremy L. Martin, Molly Maxwell, Victor Reiner, Scott O. Wilson · Journal of Combinatorics · 2015
The pseudodeterminant pdet(M ) of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant.If ∂ is a symmetric or skew-symmetric matrix then pdet(∂∂ t ) = pdet(∂) 2 .Whenever ∂ is the k th boundary map of a self-dual CW-complex 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 -1)-trees, complementing the analogous result for even-dimensional spheres given by the second author.The argument relies on the topological fact that any self-dual evendimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.