Euler totient of subfactor planar algebras
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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.References
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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
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3856145