Heisenberg SICs, Stark Units and Weil Representations

Abstract: SICs, also known as equiangular tight frames, are configurations of $d^2$ equiangular lines in $\mathbb{C}^d$ 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 $\mathcal{H}(\mathbb{Z}/d\mathbb{Z})$ via its Schr\¨odinger 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 $\mathrm{SL}_2(\mathbb{Z}/d\mathbb{Z})$ on the set of Heisenberg SICs via its $d$-dimensional Weil representation as a subgroup of the automorphism group of $\mathcal{H}(\mathbb{Z}/d\mathbb{Z})$. 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 $\mathbb{Q}(\sqrt{(d−1)(d+3)})$. On a more specialised note and as time permits, I will outline my recent work on the lifted Weil representation in the case $d = p^n$ and possible connections to a $p$-adic theory of SICs.

K-theory and homologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

Sheffield Number Theory Seminar

Series comments: Unless stated otherwise, talks will be in J11 Hicks Building. We will announce individual links for online talks.