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Euler totient of subfactor planar algebras


Author: Sébastien Palcoux
Journal: Proc. Amer. Math. Soc. 146 (2018), 4775-4786
MSC (2010): Primary 46L37; Secondary 05E10, 05E15, 06B15, 20C15, 20D30
DOI: https://doi.org/10.1090/proc/14167
Published electronically: July 23, 2018
MathSciNet review: 3856145
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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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Additional Information

Sébastien Palcoux
Affiliation: Institute of Mathematical Sciences, Chennai, India
Address at time of publication: Harish-Chandra Research Institute, Allahabad, Uttar Pradesh, 211019 India
Email: sebastienpalcoux@gmail.com

DOI: https://doi.org/10.1090/proc/14167
Keywords: Von Neumann algebra, subfactor, planar algebra, biprojection, lattice, M\"obius function, Euler totient, Boolean algebra
Received by editor(s): January 19, 2018
Received by editor(s) in revised form: February 21, 2018
Published electronically: July 23, 2018
Additional Notes: This work was supported by EPSRC grant no EP/K032208/1.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2018 American Mathematical Society