Title: Simplicial spanning trees and simplicial matrix-tree theorems
Abstract: The classical matrix-tree theorem expresses the number of spanning trees of a graph in terms of its associated Laplacian matrix. Building on the work of Bolker, Kalai and Adin, we generalize the matrix-tree theorem from graphs to the wider setting of simplicial complexes. Having defined simplicial spanning trees appropriately, we obtain a more general version of the matrix-tree theorem, expressing an analogous invariant of a simplicial complex in terms of its simplicial Laplacian matrices. The theorem holds for any pure simplicial complex that has the homology type of a wedge of spheres. Further, by assigning indeterminates to the faces of the simplicial complex, we establish a weighted version of the simplicial matrix-tree theorem, yielding more finely weighted enumerators for simplicial spanning trees (akin to the Cayley-Prufer theorem enumerating spanning trees of the complete graph by degree sequence). In the special case of a shifted complex, we give a complete description of the factors of the tree enumerator under a very fine weighting. This generalizes a result of Remmel and Williamson on threshold graphs (obtained independently by Martin and Reiner) as well as Duval and Reiner's formula for the unweighted Laplacian eigenvalues. We further conjecture that the tree enumerators of matroid complexes and color-shifted complexes admit similarly nice factorizations. This is joint work with Art Duval (University of Texas at El Paso) and Caroline Klivans (University of Chicago).