On the non-occurrence of the coxeter graphs β_2n+1, D_2n+1 and E₇ as the principal graph of an inclusion of II₁ factors

Abstract

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 β_2n+1 D_2n+1 and E₇ do not arise as Jones' principal graph invariant of an inclusion of II₁ factors. (Here, we use the symbol an to denote the graph that is normally denoted by B_n, the reason for this changed terminology being spelt out in the text.)