Big categorification on towers of classical groups and wreath product groups

Abstract

We develop a uniform framework for ``big'' categorification of representation categories of towers of finite classical groups and wreath product groups. We construct actions of symmetric products of Heisenberg categories, quantum in the finite classical group case and degenerate in the wreath product case. For modular coefficients, our standing assumptions are $\ell\nmid q(q-1)$ for the finite-classical towers and $\ell\nmid |H|$ for wreath products. These actions lead, after decomposition by the colored dot spectra over a field, to categorical actions of Kac--Moody 2-categories attached to the corresponding disjoint unions of type~A quivers, and hence to actions of large Lie algebras on Grothendieck groups. In characteristic zero, the resulting actions control the centers in every rank through diagrammatic central elements, and the associated colored weight functions separate all irreducible ordinary characters. We also obtain modular block descriptions for wreath product groups, under the standing cross-characteristic assumption, through categorification.