2015/03/01 by Guus Regts, Regts, Guus, Alexander Schrijver +3
Mathematics · #05C25 #15A72 #17Bxx #20C30 #57M27 #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.CO #math.QA #math.RT #msc:05C25 #msc:15A72 #msc:17Bxx #msc:20C30 #msc:57M27
paper · pdf · doi:10.48550/arxiv.1503.00337
arxiv created 2016/08/01 · arxiv updated 2016/08/02
A \em cyclic graph is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207--227). They are characterized by `weak reflection positivity', which amounts to the positive semidefiniteness of matrices based on the `k-join' of cubic cyclic graphs (for all k∈\oZ+). Basic tools are the representation theory of the symmetric group and geometric invariant theory, in particular the Hanlon-Wales theorem on the decomposition of Brauer algebras and the Procesi-Schwarz theorem on inequalities defining orbit spaces.