Related papers: Highest weight sl_2-categorifications I: crystals
We study equivalences for category O_p of the rational Cherednik algebras H_p of type G_l(n) = \mu_l^n\rtimes S_n: a highest weight equivalence between O_p and O_{\sigma(p)} for \sigma\in S_l and an action of S_l on a non-empty Zariski open…
Varagnolo and Vasserot conjectured an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a truncation of an affine parabolic category O of type A. In this paper we reduce their conjecture to some purely…
This paper continues the study of highest weight categorical sl_2-actions started in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the…
Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight…
We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism between a highest weight crystal and the tensor product of a perfect crystal and another highest weight crystal, all in level 1 type A affine. The nodes of the…
Highest weight categories are described in terms of standard objects and recollements of abelian categories, working over an arbitrary commutative base ring. Then the highest weight structure for categories of strict polynomial functors is…
An action of the $\mathfrak{sl}_2$-crystal category on graded/mixed (integral) category $\mathcal{O}$ `lifting' the usual tensor product is defined.
Highest weight categories are an abstraction of the representation theory of semisimple Lie algebras introduced by Cline, Parshall and Scott in the late 1980s. There are by now many characterisations of when an abelian category is highest…
We classify blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n) by using the "residue equivalence" for multi-partitions.
We show that the wall crossing bijections between simples of the category O of the rational Cherednik algebras reduce to particular crystal isomorphisms which can be computed by a simple combinatorial procedure on multipartitions of fixed…
In this survey article we review Kac-Moody and Heisenberg algebra actions on the categories $\mathcal{O}$ of the rational Cherednik algebras associated to groups $G(\ell,1,n)$. Using these actions we solve basic representation theoretic…
We introduce the representation category $\mathscr{C}({\bf G})$ for a connected reductive algebraic group ${\bf G}$ which is defined over a finite field $\mathbb{F}_q$ of $q$ elements. We show that this category has many good properties for…
In this paper we describe the characters of irreducible objects in category O for the rational Cherednik algebra associated to GL_2(F_p) over an algebraically closed field of positive characteristic p, for any value of the parameter t and…
We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight $ 2 \omega_k. $
We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm…
We categorify representations of quantum sl(n) whose highest weight is twice a fundamental weight.
We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases…
Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups $GL(X)$ in the Verlinde category $Ver_p$ in characteristic $p$. The main question we answer is how to…
Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two…
We first consider the rational Cherednik algebra corresponding to the action of a finite group on a complex variety, as defined by Etingof. We define a category of representations of this algebra which is analogous to "category O" for the…