Trees, skeleta, and combinatorics of monomial chip-firing idealsAlgebra & Discrete Mathematics
|Anton Dochtermann, Texas State University
|Mon, Apr 10 2017, 4:10PM
For any graph G, one can construct the ‘G-parking function ideal’ M_G, a monomial ideal whose standard monomials are in bijection with the spanning trees of G (and hence connect to various other combinatorial objects). Postnikov and Shapiro studied the ideals M_G in connection with power ideals and other deformations of ‘monotone monomial ideals’, and constructed minimal resolutions for certain classes. Minimal cellular resolutions of MG for arbitrary G were later described by Dochtermann and Sanyal.
The ideals M_G are also strongly related to ‘chip-firing’ on the graph G, a dynamical system on the vertices governed by the Laplacian matrix. Motivated by these notions we study certain ‘skeleta’ of the ideals MG, generated by certain subsets of the vertices of G. For some large classes we construct minimal resolutions and describe monomial bases. These constructions involve a number of combinatorial gadgets including tropical hyperplanes, the ‘signless’ Laplacian, and (new?) enumerations of Cayley trees.