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

TL;DR

This paper categorifies the Jacobi-Trudi formula using quasi-hereditary quotients of cyclotomic KLR algebras, revealing their nil-Koszul structure and connections to BGG resolutions and Koszul duality.

Contribution

It introduces Jacobi-Trudi algebras as a new class of nil-Koszul algebras arising in categorification, linking BGG resolutions with Koszul duality.

Findings

Dominant simple modules admit BGG resolutions by permutation modules.

Jacobi-Trudi algebras are shown to be nil-Koszul.

Koszul duality recovers differentials of BGG resolutions.

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.