Vol. 292, No. 1, 2018

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ISSN: 0030-8730
Ore's theorem on cyclic subfactor planar algebras and beyond

Sebastien Palcoux

Vol. 292 (2018), No. 1, 203–221
Abstract

Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore’s theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that this is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a nontrivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.

Keywords
von Neumann algebra, subfactor, planar algebra, biprojection, distributive lattice, finite group, representation
Mathematical Subject Classification 2010
Primary: 46L37
Secondary: 06D10, 16W30, 20C15, 20D30
Milestones
Received: 5 February 2017
Revised: 26 July 2017
Accepted: 2 August 2017
Published: 22 September 2017
Authors
Sebastien Palcoux
Institute of Mathematical Sciences
Chennai 600113
India