Euler totient of subfactor planar algebras

Palcoux, Sébastien

doi:10.1090/proc/14167

Euler totient of subfactor planar algebras
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Proc. Amer. Math. Soc. 146 (2018), 4775-4786 Request permission

Abstract:

We extend Euler’s totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Möbius 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.

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