Cyclic symmetries of linear systems on graphs
In the divisor theory of graphs, a finite, connected graph is viewed as a discrete analog of a Riemann surface. A divisor D on a graph is an assignment of integers to each vertex of the graph. An important statistic in this setting is the complete linear system of D, which is the collection of effective (e.g. non-negative) divisors linearly equivalent to D via the discrete Laplacian operator. When the underlying graph for divisors is the cycle graph, we present a poset isomorphism between a subset of effective divisors under the firing operator and a subposet of Young’s lattice defined by R. Suter, H. Thomas and N. Williams. Using the combinatorics of these divisors, we give an alternate proof of a special case of the result by H. Thomas and N. Williams that the Suter's poset exhibits the cyclic sieving phenomenon (CSP), which was introduced by V. Reiner, D. Stanton and D. White.