Möbius Strip Diagram Algebras
D. W. Collison, D. Tubbenhauer
TL;DR
The paper introduces Möbius strip diagram algebras as a nonorientable extension of partition diagram calculi, realized as linear quotients of a nonorientable $2$D cobordism category, and develops their algebraic and representation-theoretic structure. By constructing explicit normal forms and a cobordism-driven identification, it connects these algebras to a rich TQFT-inspired framework and Deligne-type modular decompositions, while establishing a robust sandwich cellular theory to classify simples via H-reduction. The work provides comprehensive classification of simple modules through apex data and monoid wreath products, supported by Gram-matrix techniques that yield explicit dimension formulas and semisimplicity criteria. Together, these results illuminate the interplay between nonorientable topology, diagrammatic algebras, and modular representation theory, with concrete combinatorial counts and examples illustrating the impact of Möbius features on classical partition-style answers.
Abstract
We introduce Möbius strip diagram algebras (and their monoid and categorical versions) as subalgebras of a partition-style diagram calculus in which strands may carry handles and Möbius strip features. We identify the resulting diagram category with a linear quotient of a nonorientable two-dimensional cobordism category. Finally, we develop the associated cell theory and use it to classify the simple modules and compute dimensions in a range of cases.