On uniqueness of tensor products of irreducible categorifications

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.

arXiv commentv3: updated with to fix some conventional issues; E and F have swapped roles in the final section

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• Semi-infinite highest weight categories
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• On categories O for quantized symplectic resolutions
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• Highest weight categories and Macdonald polynomials
• Isotypic faithful $2$-representations of $\mathcal{J}$-simple fiat $2$-categories
• Highest weight theory for finite-dimensional graded algebras with triangular decomposition
• Highest weight sl_2-categorifications II: structure theory
• p-DG cyclotomic nilHecke algebras
• Homological invariants in category O for the general linear superalgebra
• Categorification of tensor product representations of sl(k) and category O
• Categorification via blocks of modular representations for sl(n)
• On modular categories O for quantized symplectic resolutions
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Categorical actions and multiplicities in the Deligne category $\underline{Rep}(G{L}_t)$
• Representations of weakly triangular categories
• Representations of cyclotomic oriented Brauer categories
• Categorification of infinite-dimensional ${\mathfrak{sl}}_2$-modules and braid group 2-actions I: tensor products
• Graded triangular bases