Three perspectives on categorical symmetric Howe duality

Abstract In this article, we consider the categorical symmetric Howe duality introduced by Khovanov, Lauda, Sussan, and Yonezawa. While originally defined from a purely diagrammatic perspective, this construction also has geometric and representation-theoretic interpretations, corresponding to certain perverse sheaves on spaces of quiver representations and the category of Gelfand–Tsetlin modules over $\mathfrak {gl}_n$ . In particular, we show that the “deformed Webster algebras” discussed in [KLSY18] manifest a Koszul duality between blocks of the category of Gelfand–Tsetlin modules over $\mathfrak {gl}_n$ , and the constructible sheaves on representations of a linear quiver invariant under a certain parabolic in the group that acts by changing bases. Furthermore, we show that this duality intertwines translation functors with a diagrammatic categorical action (generalizing that of [KLSY18]).