Boris Shapiro: On some algebras associated with a simple graph

Around 2000 A. Postnikov and myself introduced two pairs of commutative algebras associated with any simple graph G. One subalgebra in each pair is associated with a monomial ideal and another with an ideal generated by powers of linear forms. Both algebras in each pair have the same Hilbert series which in the first case is the generating function of the number of spanning forests and in the second case is the generating function of the number of spanning trees in G according to their external activity. These Hilbert series can be obtained as specializations of the Tutte polynomial of G. In connection with this project we also introduced the notion of a G-parking function which is now actively studied by specialists in combinatorics. Recently joint with A. Kirillov we defined two new algebras which can be thought as a ‘K-theoretic’ analogue of the former algebras. They seem to have the same total dimensions (i.e. the total number of forests and trees resp.) but according to some new statistics which is still a mystery. I will report on this work in progress, formulate some few results and state several conjectures.