On representations of Clifford algebras of ternary cubic forms

TL;DR

This paper explores the relationship between Clifford algebra representations of ternary cubic forms and Ulrich bundles on cubic surfaces, establishing the existence of irreducible representations of all dimensions.

Contribution

It introduces a correspondence between Clifford algebra representations and Ulrich bundles, and proves the existence of irreducible representations of all possible dimensions.

Findings

Established a one-to-one correspondence between algebra representations and vector bundles.

Proved the existence of irreducible representations of every possible dimension.

Utilized recent classification results to support the findings.

Abstract

In this article, we provide an overview of a one-to-one correspondence between representations of the generalized Clifford algebra $C_f$ of a ternary cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic surface $X$. We study general properties of Ulrich bundles, and using a recent classification of Casanellas and Hartshorne, deduce the existence of irreducible representations of $C_f$ of every possible dimension.