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After discussing some preliminaries on the notion of an action of a hypergroup on a set, we present elementary proofs of the fact that the Coxeter graphs /?2«+i > D2n+ι and EΊ do not arise as Jones' principal graph invariant of an inclusion of Hi factors. (Here, we use the symbol βn to denote the graph that is normally denoted by Bn, the reason for this changed terminology being spelt out in the text.) In this paper, we define and discuss some elementary consequences of the notion of an action of a hypergroup on a set and go on to use this notion to provide an elementary proof of the fact that the Coxeter graphs βin+\ > £>2«+i and EΊ do not arise as Jones' principal graph invariant of an inclusion of II i factors. (The symbol βn , rather than the symbol Bn , is used here to denote the graph

pacific journal of mathematics

On the nonoccurrence of the Coxeter graphsβ2n+1, D2n+1andE7as the principal graph of an inclusion of II1factors