On uniqueness of tensor products of irreducible categorifications
Ivan Losev, Ben Webster
TL;DR
The paper introduces an axiomatic framework for tensor product categorifications of tensor products of irreducible g-modules and establishes a strong uniqueness theorem: any such categorification is strongly equivariantly equivalent to the canonical model C(ν̲). It also proves that the associated simple objects carry a crystal structure that is isomorphic to the tensor-product crystal B(ν_1)⊗…⊗B(ν_n), and provides a detailed splitting and double centralizer analysis to realize this equivalence. Concrete realizations include diagrammatic T^{ν} algebras and cyclotomic q-Schur quotients in type A, as well as parabolic category O constructions, linking categorification to classical representation theory. The results yield a robust, universal picture for tensor-product categorifications, with implications for graded lifts, internal tensor products, and connections to Lie superalgebras via subsequent work.
Abstract
In this paper, we propose an axiomatic definition for a tensor product categorification. A tensor product categorification is an abelian category with a categorical action of a Kac-Moody algebra g in the sense of Rouquier or Khovanov-Lauda whose Grothendieck group is isomorphic to a tensor product of simple modules. However, we require a much stronger structure than a mere isomorphism of representations; most importantly, each such categorical representation must have standardly stratified structure compatible with the categorification functors, and with combinatorics matching those of the tensor product. With these stronger conditions, we recover a uniqueness theorem similar in flavor to that of Rouquier for categorifications of simple modules. Furthermore, we already know of an example of such a categorification: the representations of algebras T^λpreviously defined by the second author using generators and relations. Next, we show that tensor product categorifications give a categorical realization of tensor product crystals analogous to that for simple crystals given by cyclotomic quotients of KLR algebras. Examples of such categories are also readily found in more classical representation theory; for finite and affine type A, tensor product categorifications can be realized as quotients of the representation categories of cyclotomic q-Schur algebras.