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Sebastien Palcoux

Publications and source records attributed to Sebastien Palcoux.

26 records · Page 2Linked to original sources
arXiv2018

Euler totient of subfactor planar algebras

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a minimal 2-box projection generating the identity biprojection. We explain a relation with a problem of K.S. Brown. As an application, we define the dual Euler totient of a finite group and we show that if it is nonzero then the group admits a faithful irreducible complex representation. We also get an analogous result at depth 2, involving the central biprojection lattice.

math.OA↗
arXiv2018

On Boolean intervals of finite groups

We prove a dual version of Øystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient $\hatφ$. For any Boolean group-complemented interval, we observe that $\hatφ = φ\neq 0$ by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that $\hatφ$ is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval $[H,G]$, the graded coset poset $\hat{P} = \hat{C}(H,G)$ is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex $Δ(P)$ is $\hatφ$, so nonzero. We deduce that these results are true beyond the group-complemented case with $|G:H|<32$. One observes that they are also true when $H$ is a Borel subgroup of $G$.

math.GR↗
arXiv2017

Ore's theorem on cyclic subfactor planar algebras and beyond

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.

math.OA↗
arXiv2016

Spectral triples for finitely generated groups, index 0

Using Cayley graphs and Clifford algebras, we are able to give, for every finitely generated groups, a uniform construction of spectral triples with a generically non-trivial phase for the Dirac operator. Unfortunatly $D_{+}$ is index $0$, but we are naturally led to an interesting classification of finitely generated groups into three types.

math.OA↗
arXiv2016

The type III manufactory

Using unusual objects in the theory of von Neumann algebra, as the chinese game Go or the Conway game of life (generalized on finitely presented groups), we are able to build, by hands, many type III factors.

math.OA↗
arXiv2016

Dual Ore's theorem for distributive intervals of small index

This paper proves a dual version of a theorem of Oystein Ore for every distributive interval of finite groups [H,G] of index |G:H|<9720, and for every boolean interval of rank <7. It has applications to representation theory for every finite group.

math.GR↗
arXiv2016

Ore's theorem for cyclic subfactor planar algebras and applications

This paper introduces the cyclic subfactors, generalizing the cyclic groups as the subfactors generalize the groups, and generalizing the natural numbers as the maximal subfactors generalize the prime numbers. On one hand, a theorem of O. Ore states that a finite group is cyclic if and only if its subgroups lattice is distributive, and on the other hand, every subgroup of a cyclic group is normal. Then, a subfactor planar algebra is called cyclic if all the biprojections are normal and form a distributive lattice. The main result shows in what sense a cyclic subfactor is singly generated, by generalizing one side of Ore's theorem as follows: if a subfactor planar algebra is cyclic then it is weakly cyclic (or w-cyclic), i.e. there is a minimal 2-box projection generating the identity biprojection. Some extensions of this result are discussed, and some applications of it are given in subfactors, quantum groups and finite group theories, for example, a non-trivial upper bound for the minimal number of irreducible complex representations generating the left regular representation.

math.OA↗