Ore's theorem on cyclic subfactor planar algebras and beyond

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 it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.

20 pages (it is a short version of arXiv:1505.06649). To appear in Pacific J. Math

References in corpus (1)

• Ore's theorem for cyclic subfactor planar algebras and applications

Cited by in corpus (7)

• Fusion Bialgebras and Fourier Analysis
• Interpolated family of non group-like simple integral fusion rings of Lie type
• On Boolean intervals of finite groups
• Ore's theorem on subfactor planar algebras
• Euler totient of subfactor planar algebras
• An angle between intermediate subfactors and its rigidity
• Boolean lattices in finite alternating and symmetric groups