Heisenberg SICs, Stark Units and Weil Representations

Abstract: SICs, also known as equiangular tight frames, are configurations of d² equiangular lines in ℂᵈ whose applications in signal processing and quantum physics have been known and studied for more than 30 years. More recently, numerical investigations of so-called Heisenberg SICs (which have an action of 𝓗(ℤ/dℤ) via its Schrödinger representation) have revealed surprising, heuristic connections with conjectural 'Stark units' and hence with Hilbert's 12th Problem over real-quadratic fields.

Just as intriguingly, the action of Galois on these units seems to be connected to the action of SL₂(ℤ/dℤ) on the set of Heisenberg SICs via its d-dimensional Weil representation as a subgroup of the automorphism group of 𝓗(ℤ/dℤ).

In my talk, I will first give an overview of recent SIC-related research, as well as the Stark Conjectures (which date from the 1970s but are still largely unproven). I will explain the experimental evidence connecting SICs with Stark-Units over the field ℚ(√(d − 1)(d + 3)). On a more specialised note and as time permits, I will outline my recent work on the lifted Weil representation in the case d = pⁿ and possible connections to a p-adic theory of SICs.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.