On category O for affine Grassmannian slices and categorified tensor products

Abstract

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra frakturg, which we will assume is simply laced. In this paper, we relate the category scriptO over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category scriptO is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category scriptO for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category scriptO are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U(frakturgln) and its associated W‐algebras.

Representations of shifted quantum affine algebras

“…Nakajima explained to the author this bijection preserves finiteness of dimension and category O. Thus, combining with [KTWWY2], this gives an explicit parametrization of simple representations in category O of truncated non simply-laced shifted Yangians and quantum affine algebras. After using the comparison between simply-laced and twisted q-characters [He4], one can consider a possible relation between the two parametrizations.…”

“…Up to isomorphism, U µ + ,µ − q (ĝ) only depends on µ and will be denoted simply by U µ q (ĝ). For simply-laced types, representations of shifted Yangians and related Coulomb branches have been intensively studied, see [BrK,KTWWY1,KTWWY2] and references therein. For non simply-laced types, representations of quantizations of Coulomb branches have been studied by Nakajima and Weekes [NW] by using the method originally developed in [N5] for simply-laced types.…”

“…Let us comment on related structures and on previous results. For simply-laced types, simple representations of truncated shifted Yangians have been parametrized in terms of Nakajima monomial crystals [KTWWY2]. Combining with [N5], this implies an analog statement for simply-laced shifted quantum affine algebras.…”

Shifted Quantum Affine Algebras: Integral Forms in Type A

“…The following result is due to [KTWWY,Corollary 4.10] (see Remark 4.33 for an alternative proof, based on the shuffle realizations of Y (sl n ), Y (sl n ) of [T, §6]):…”

Perfect bases in representation theory: three mountains and their springs

“…In particular, following the philosophy of Braden-Licata-Proudfoot-Webster [11], the MV cycles and quiver variety components can be categorified using category O for quantizations of affine Grassmannian slices and quiver varieties, respectively. Moreover, these categories are closely related to categories of modules for KLR algebras [28]. From this perspective, the failures of these bases to agree with the dual canonical basis in Theorem 1.13 can be attributed to the non-irreducibility of the character varieties of simple objects in these categories.…”