Representations of the oriented skein category

Abstract

The oriented skein category OS(z, t) is a ribbon category which underpins the definition of the HOMFLY-PT invariant of an oriented link, in the same way that the Temperley-Lieb category underpins the Jones polynomial. In this article, we develop its representation theory using a highest weight theory approach. This allows us to determine the Grothendieck ring of its additive Karoubi envelope for all possible choices of parameters, including the (already well-known) semisimple case, and all non-semisimple situations. Then we construct a graded lift of OS(z, t) by realizing it as a 2-representation of a Kac-Moody 2-category. We also discuss the degenerate analog of OS(z, t), which is the oriented Brauer category OB(δ).

Semi-infinite highest weight categories

“…The conjecture is also consistent with the so-called Feigin-Frenkel conjecture [AF1,Conjecture 4.7], which says that composition multiplicities of restricted Verma modules are related to the periodic Kazhdan-Lusztig polynomials from [Lus] (and Jantzen's generic decomposition patterns from [Jan2]). These polynomials depend on the relative position of the given pair of weights and, product categorifications; see [Bru,Theorem 1.11]. We just note that ReppGL δ q is the additive Karoubi envelope of the oriented Brauer category OBpδq for δ P C, which was mentioned already in the previous subsection.…”

Semisimplification of the category of tilting modules for GL_n

“…sending ↑ to the natural G n -module V n and ↓ to the dual module V * n . By a version of Schur-Weyl duality, this functor is full, and it is dense if either p = 0 or p > n; e.g., see [B,Theorem 1.10].…”

“…When p = 0 or p > n (and t is the image of n in k still), the functor Ψ n induces an equivalence of symmetric monoidal categories between Kar(OB(t)/I n ) and Tilt(G n ), where I n is the tensor ideal of OB(t) generated by the endomorphism of ↑ ⊗(n+1) associated to the quasi-idempotent g∈Sn+1 (−1) ℓ(g) g in the group algebra kS n+1 of the symmetric group. This is explained in detail in [B,Theorem 1.10].…”

String diagrams and categorification

“…The quantum analogue of the oriented Brauer category is the HOMFLY-PT skein category introduced by Turaev in [Tur89, §5.2], where it was called the Hecke category. See also [Bru17,BSW18b].…”