Cubist Algebras

TL;DR

This paper introduces 'Cubist algebras' derived from rhombohedral tilings, revealing their homological properties, derived equivalences, and connections to symmetric group blocks, advancing algebraic and combinatorial understanding.

Contribution

It constructs Cubist algebras from tilings, proves their homological properties, and links them to symmetric group representation theory, providing new algebraic structures and equivalences.

Findings

Cubist algebras satisfy Koszulity and quasi-heredity.

Derived equivalences relate different tiling configurations.

Connections established between Cubist algebras and symmetric group blocks.

Abstract

We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these `Cubist algebras' satisfy strong homological properties, such as Koszulity and quasi-heredity, reflecting the combinatorics of the tilings. We construct derived equivalences between Cubist algebras associated to local mutations in tilings. We recover as a special case the Rhombal algebras of Michael Peach and make a precise connection to weight 2 blocks of symmetric groups.