For relational structures A, B, the Promise Constraint Satisfaction Problem PCSP(A, B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. Note that if there exists C with homomorphisms A ? C ? B, then PCSP(A, B) reduces to CSP(C). All known tractable PCSPs seem to reduce to tractable CSPs in this way. However Barto (2019) showed that some PCSPs over finite structures require solving CSPs over infinite C. We provide examples showing that even when a reduction to finite C is possible, this structure may become arbitrarily large. This is joint work with Alexandr Kazda and Dmitriy Zhuk.
Khovanov-Lauda-Rouquier (KLR) algebras arise in the categorification theory of quantum groups. In affine type A, KLR representations can be studied through the lens of cuspidal modules, which categorify root vectors in PBW bases of the associated quantum group, or through the lens of Specht modules, associated with the cellular structure of cyclotomic KLR algebras. Cuspidal ribbons provide a sort of combinatorial bridge between these approaches. I will describe the combinatorics of cuspidal ribbon tableaux, as well as some implications they have in the world of KLR representation theory, such as providing bounding information on simple factors of Specht modules, as well as presentations of cuspidal modules.
Cuspidal ribbon tableaux and representations of KLR algebras Sponsored by the Meyer Fund
Solving small PCSPs via large CSPs