Séminaire Lotharingien de Combinatoire, 78B.84 (2017), 12 pp.

Marcelo Aguiar and Swee Hong Chan

Toric Arrangements Associated to Graphs

Abstract. We study certain toric arrangements associated to graphs. The arrangement depends on the choice of an integral lattice: we focus on the case of the (co)root lattice of type A, but also comment on the (simpler) case of the (co)weight lattice of type A. We obtain a combinatorial description for the intersection poset and derive several results on the characteristic polynomial and the arithmetic Tutte polynomial of the toric arrangement. The former counts proper colorings that satisfy an additional divisibility condition. By employing the Voronoi cell of the lattice, we show that the chambers of certain toric arrangements may be seen as equivalence classes for a canonical equivalence relation on the set of chambers of the corresponding linear arrangement. We study this relation in the graphic case.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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