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We define weak 2‐categories of finite‐dimensional algebras with bimodules, along with collections of operators O(c, x) on these 2‐categories. We prove that special examples Op of these operators control all homological aspects of the rational representation theory of the algebraic group GL2, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators O(c, x) honour derived equivalences in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight ℤ+‐gradings on Schur algebras S(2, r), and the existence of braid group actions on the derived categories of blocks of these Schur algebras.

Abstract. Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra C ← − p (F), with a transparent combinatorial structure, which controls the rational representation theory of GL 2 (F).

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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.