Discrete Invariants of Koszul Artin-Schelter Regular Algebras of Dimension four
Vishal Bhatoy, Colin Ingalls
TL;DR
The paper develops a computational, representation-theoretic pipeline to classify 4-dimensional Koszul Artin–Schelter regular algebras by constructing their defining superpotentials in $V^{\otimes 4}$, decomposing the invariants via Schur–Weyl duality, and interpreting them as sections on partial flag varieties through the Borel–Weil theorem. By computing geometric invariants $X_{\lambda}(w)$ on flag varieties for partitions $\lambda\vdash 4$ (notably $(4),(3,1),(2,2),(2,1,1)$ and $(1,1,1,1)$) across 77 known families with Magma, the authors obtain discrete invariants that distinguish algebras, including cases where the loci are K3-like or reducible quartics. They assemble these invariants into tables for the four partitions, discuss absolute irreducibility via base change, and use them to partition the algebraic stack $\mathcal{A}_4$ into boxes, proving that generic algebras from different boxes are non-isomorphic. The work advances noncommutative projective geometry by providing explicit, computable invariants that separate AS-regular fourfolds beyond twist-equivalence and clarifies the geometric meaning of these invariants in terms of flag-variety sections. Significance lies in a concrete, geometry-grounded approach to distinguishing high-dimensional AS-regular algebras with potential connections to rich geometric structures such as quartic surfaces and K3-type invariants.
Abstract
We compute the superpotentials for known families of Koszul Artin-Schelter regular algebras of dimension four using Magma, and apply Schur-Weyl duality from representation theory to determine the relevant invariants. Through the Borel-Weil theorem, we interpret these invariants as sections of line bundles over partial flag varieties, resulting in geometric invariants that, in some cases, correspond to K3 surfaces. We compute discrete invariants of these geometric invariants and use them to distinguish algebras.