Cohomological integrality isomorphisms

The theory of cohomological Hall algebras has proven powerful in studying Donaldson-Thomas invariants of some 3-Calabi-Yau categories. It is in particular crucial to obtain cohomological integrality identities. Roughly speaking, the cohomological integrality results concern the finiteness of some invariants associated to the category. They also give cohomologically refined invariants. In my talk, I will give a brief overview of such results, explain their meaning, and explain how to obtain them in various contexts. I will concentrate on a new situation given by symmetric representations of reductive groups.