We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.
A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi lectures given by the author in Hokkaido University in June 2009.
We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.
We give a graphical calculus for a categorification of a Clifford algebra and its Fock space representation via differential graded categories. The categorical action is motivated by the gluing action between the contact categories of infinite strips.
The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the cell modules of the Temperley-Lieb algebra.
In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra. One of our primary results is that these categories coincide when both are defined. We also inves…
Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise naturally in geometric representation theory and in modular representation theory of symmetric groups. A framework for studying categorical representations was introduced by Rouquier and Khovanov-Lauda. Their definitions are algebraic/diagrammatic, but are connected to the topology of quiver varieties by the work of Rouquier and Varagnolo-Vasserot. In this …
We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac--Moody algebras, and in particular of parabolic 2-Verma module.
In this expository paper we present an overview of various graphical categorifications of the Heisenberg algebra and its Fock space representation. We begin with a discussion of "weak" categorifications via modules for Hecke algebras and "geometrizations" in terms of the cohomology of the Hilbert scheme. We then turn our attention to more recent "strong" categorifications involving planar diagrammatics and derived categories of coherent sheaves on Hilbert schemes.
In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N greater than or equal to 2, with a view towards categorifying the corresponding torus skein algebras via Khovanov-Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subal…
We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category $\mathcal{O}$; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the re…
We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.
Symmetric functions appear in many areas of mathematics and physics, including enumerative combinatorics, the representation theory of symmetric groups, statistical mechanics, and the quantum statistics of ideal gases. In the commutative (or "even") case of these symmetric functions, Kostant and Kumar introduced a nilHecke algebra that categorifies the quantum group $U_q(\mathfrak{sl}_2)$. This categorification helps to better understand Khovanov homology, which has important applications in studying knot polynomials and gauge theory. Recently, Ellis and Khovanov initiated the program of "oddi…
