Pseudodeterminants and perfect square spanning tree counts

Abstract

The pseudodeterminant $\text{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then $\text{pdet}(\partial {\partial }^t)=\text{pdet}(\partial {)}^2$. Whenever $\partial$ 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 \emph{antipodally} self-dual CW-sphere of odd dimension, the pseudodeterminant of its $k$th 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 even-dimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.

Final version; minor revisions. To appear in Journal of Combinatorics

References in corpus (2)

• Kirchhoff's theorems in higher dimensions and Reidemeister torsion
• On the Tutte-Krushkal-Renardy polynomial for cell complexes