BGG resolutions, Koszulity, and stratifications, part II: the Jacobi-Trudi algebra

Abstract

We categorify the Jacobi-Trudi determinant formula for Schur functions as a shadow of a highest-weight phenomenon by considering certain quasi-hereditary quotients of certain cyclotomic KLR algebras, which we call ``Jacobi-Trudi algebras''. These algebras come equipped with a map from $\mathbb{C} S_n$, and we show that the dominant simple modules for these algebras admit BGG resolutions which, when restricted to $\mathbb{C} S_n$, become resolutions of Specht modules by permutation modules. We establish these BGG resolutions by showing that these Jacobi-Trudi algebras, as well as the Soergel calculi to which they are Morita equivalent, are ``nil-Koszul'', meaning that they have ``lower half subalgebras'' which are Koszul. We also show that Koszul duality with respect to this half subalgebra can be used to recover the differentials of the BGG resolutions. Hence this paper gives another example of a nil-Koszul algebra appearing naturally in categorification and gives another demonstration of the intricate connection between nil-Koszulity and BGG resolutions.