SICs and the Triangle Group (3, 3, 3)

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Author: Yakymenko, Danylo
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212263
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: SICs and the Triangle Group (3, 3, 3). Danylo Yakymenko. SIGMA 20 (2024), 044, 12 pages

The problem of the existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group (3, 3, 3). We give a full description of such representations and show how they can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension > 3 is prime.