Pseudodeterminants and perfect square spanning tree counts

The pseudodeterminant $\mathrm{pdet}(M)$ 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 $\mathrm{pdet}({\partial \partial}^t) = \mathrm{pdet}(\partial)^2$. Whenever $\partial$ is the $k^\textrm{th}$ 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 $k^\textrm{th}$ 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.

Keywords

pseudodeterminant, spanning tree, Laplacian, Dirac operator, perfect square, central reflex, self-dual

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Published 4 June 2015