CATEGORIFICATION OF SOME LEVEL TWO REPRESENTATIONS OF QUANTUM 𝔰𝔩_n

Abstract

We categorify representations of quantum 𝔰𝔩n whose highest weight is twice a fundamental weight.

$${\mathfrak {sl}}_3$$ sl 3 -Web bases, intermediate crystal bases and categorification

“…For example the following tableau is in Std( λ = ((3, 2, 1), (2, 1), (3, 2, 1))). In order to give a growth algorithm for webs with flow lines u f ∈ B J S (that is the set of all non-elliptic webs ∂u = S with a fixed flow f that extends J), we first define an injection of the set of all non-elliptic webs with flows associated to a certain pair (S, J) (or equivalent a column-strict 27 tableau or a 3-multipartition λ) into Std( λ). Moreover, we denote the set all webs with boundary datum (S, J) and a chosen flow by W J S .…”

“…Our approach in this paper is as follows. One main ingredient is the usage of categorified, diagrammatic quantum skew Howe duality studied recently independently by varies authors in this framework [13], [39] and [44] to cite a few (the first appearance in the context of sl 2 -webs seems to be the paper of Huerfano and Khovanov [27], although they never used the notion of skew Howe duality). In the sl 3 case this means that there is a strong 2-representation Ψ : U(sl n ) → Foam 3 , called foamation, of Khovanov-Lauda's [35] categorification ofU(sl n ), denoted by U(sl n ), to the category of sl 3 -foams Foam 3 .…”

slN-web categories and categorified skew Howe duality

“…For N = 2 Huerfano and Khovanov [21] used the arc algebras to category V Λ , using a (partial) categorification of quantum skew Howe duality. The representation theory of the arc algebras and its connection with geometry have been studied in depth by a variety of people [7,8,9,10,11,17,22,24,42,43].…”

On selfadjoint functors satisfying polynomial relations

“…The main motivation for the present paper stems from the recent activities on categorification of representations of various algebras, see, in particular, [CR,FKS,HS,HK,KMS1,Lau,KL,MM,MS1,MS3,MS4,R,Zh1,Zh2], the reviews [KMS2,Ma2] and references therein. In these articles one could find several results of the following kind: given a field k, an associative k-algebra Λ with a fixed generating set {a i }, and a Λ-module M, one constructs a categorification of M, that is an abelian category C and exact endofunctors {F i } of C such that the following holds: The Grothendieck group [C] of C (with scalars extended to an appropriate field) is isomorphic to M as a vector space and the functor F i induces on [C] the action of a i on M. Typical examples of algebras, for which categorifications of certain modules are constructed, include group algebras of Weyl groups, Hecke algebras, Schur algebras and enveloping algebras of some Lie algebras.…”