We show that the triply graded Khovanov-Rozansky homology of the torus link T n,k stablizes as k → ∞. We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes Pn of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that Pn is a stable limit of Rouquier complexes. A certain derived endomorphism ring of Pn computes the aforementioned stable homology of torus links.

We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n, n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links with Hilbert schemes. CONTENTS 1 arXiv:1603.00407v1 [math.GT]

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type $A$. As an application we obtain a derived annular Khovanov–Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus.

This paper lays the groundwork for the theory of categorical diagonalization. Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each eigenspace. These idempotents are mutually orthogonal and sum to the identity. We categorify these tools. At the categorical level, one has not only eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to its eigenvalues. Given an invertible endofunctor of a triangulated category with a sufficiently nice collection of eigenmaps, we construct idempotent functors which project to eigencategories. These idempotent functors are mutually orthogonal, and a convolution thereof is isomorphic to the identity functor.In several sequels to this paper, we will use this technology to study the categorical representation theory of Hecke algebras. In particular, for Hecke algebras of type A, we will construct categorified Young symmetrizers by simultaneously diagonalizing certain functors associated to the full twist braids. CONTENTS1 Extended Introduction 4 1. Summary of the theory 4 2. Subtleties of categorical diagonalization 21 3. Applications of categorical diagonalization 27 2 Categorical Diagonalization 35 4. Convolutions and homotopy categories 35 5. Decompositions of identity 45 6. Pre-diagonalizability and diagonalizability 50 7. Interpolation complexes for invertible eigenvalues 55 8. The diagonalization theorem 62 9. Generalizations and the Casimir element 68 Appendix A. Commuting properties 73 References 81

In the Iwahori-Hecke algebra, the full twist acts on cell modules by a scalar, and the half twist acts by a scalar and an involution. A categorification of this statement, describing the action of the half and full twist Rouquier complexes on the Hecke category, was conjectured by Elias-Hogancamp, and proven in type A. In this paper we make analogous conjectures for the p-canonical basis, and the Hecke category in characteristic p. We prove the categorified conjecture in type C2, where the situation is already interesting. The decategorified conjecture is confirmed by computer in rank ≤ 6; information is found in the appendix, written by Joel Gibson. CONTENTS1. New conjectures about p-cells 1.1. Action of the half twist on the Kazhdan-Lusztig basis 1.2. Action of the half twist on the p-canonical basis 1.3. Numerics and evidence 1.4. Action of the half twist complex 1.5. Diagonalizing the full twist: characteristic zero 1.6. Diagonalizing the full twist: characteristic p 1.7. Outline and acknowledgments 2. Type C 2 diagrammatics 2.1. Setup and notation 2.2. Basics of thick calculus 2.3. Direct sum decompositions 2.4. Endomorphisms of B w 0 2.5. Morphisms involving B w 0 2.6. Left multiplication versus right multiplication 3. Type C 2 results 3.1. The half twist 3.2. Action of the half twist 3.3. The full twist 3.4. The eigenmaps 3.5. The s-eigenmap and 2-torsion 3.6. Methods of computation 4. Type C 3 4.1. p-cells and their eigenvalues 4.2. Musings on the categorification: guessing the minimal complexes 4.3. Musings on the categorification: eigenvalue swap 5. Appendix: p-cells and their eigenvalues 5.1. Rank 2 5.2. Rank 3