Sporadic SICs and exceptional Lie algebras

A while back, I had a bit of a sprawling conversation about certain geometrical oddities over multiple threads at the n-Category Café. I finally got organized enough to gather these notes together, incorporating edits for clarity and recording one construction I haven’t found written in the literature anywhere.

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy.

UPDATE (28 February 2019): Thanks to a kind offer by John Baez, we’re going through this material step-by-step over at a blog with a community, the n-Category Café:

  • Part 1: Definitions and preliminaries
  • Part 2: Qutrits and E6