2014/06/19 by Melody Chan, Chan, Melody, Darren Glass +9
Mathematics · #05C05 #05C25 #05E18 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C05 #msc:05C25 #msc:05E18
paper · pdf · doi:10.48550/arxiv.1406.5147
13 pages, 9 figures
arxiv created 2014/06/19 · arxiv updated 2014/06/20
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.