Three perspectives on categorical symmetric Howe duality
arXiv:2001.07584 · doi:10.4153/S0008414X25101867
The paper studies categorical symmetric Howe duality from diagrammatic, geometric, and representation‑theoretic viewpoints, establishing a Koszul duality between blocks of Gelfand‑Tsetlin modules for glₙ and constructible sheaves on linear quiver representations, and shows how this duality intertwines translation functors with a diagrammatic categorical action.
Abstract
In this paper, 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 . In particular, we show that the "deformed Webster algebras" discussed in work of Khovanov-Lauda-Sussan-Yonezawa manifest a Koszul duality between blocks of the category of Gelfand-Tsetlin modules over , 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. Includes an appendix by the author and Jerry Guan.
final version, 35 pages, to appear in the Canadian Journal of Mathematics