Hall monoidal categories and categorical modules

الأصل · EN

We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are simplicial groupoids satisfying the 2-Segal conditions (as introduced by Dyckerhoff and Kapranov), the main examples come from Waldhausen's S-construction. To treat the case of modules, we introduce a relative version of the 2-Segal conditions. Furthermore, we generalize a classical result about the representation theory of symmetric groups to the case of wreath product groups: We construct a monoidal equivalence between the category of complex G Sₙ-representations (for a fixed finite group G and varying n) and the category of "G-equivariant" polynomial functors; we use this equivalence to prove a version of Schur-Weyl duality for wreath products.