2017/02/28 by Sebastien Palcoux, Sébastien Palcoux · 9 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Computer science #Conjecture #Distributive lattice #Distributive property #Group (periodic table) #Lattice (music) #Mathematics #Normal subgroup #Planar #Pure mathematics #Regular representation #math.CO #math.GR #math.OA #math.QA #math.RT #msc:05E10 #msc:05E15 #msc:06D10 #msc:20C15 #msc:46L37
paper · pdf · doi:10.2140/pjm.2018.292.203
published in Pacific Journal of Mathematics 292(1), 203-221 (Mathematical Sciences Publishers) · 20 pages (it is a short version of arXiv:1505.06649). To appear in Pacific J. Math
arxiv created 2017/08/05 · arxiv updated 2017/09/28 · openalex publication_date 2017/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.