Heisenberg SICs, Stark Units and Weil Representations (David Solomon)

SICs, also known as equiangular tight frames, are configurations of d2 equian-
gular lines in Cd whose applications in signal processing and quantum physics
have been known and studied for more than 30 years. More recently, nu-
merical investigations of so-called Heisenberg SICs (which have an action of
H(Z/dZ) via its Schr¨odinger representation) have revealed surprising, heuris-
tic 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 con-
nected 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 re-
cent work on the lifted Weil representation in the case d = pn and possible
connections to a p-adic theory of SICs