Categorified Young symmetrizers and stable homology of torus links II

Abstract

We construct complexes P 1 n of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture [12] of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for P 1 n and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest. Definition 1.1. Let R → P ∨ 1 n denote an injective resolution of R, thought of as a graded R ⊗ R W R-module.We note for future reference that R ⊗ R W R is self-injective. The complex P ∨ 1 n is graded infinite, being supported in all non-negative homological degrees. This complex categorifies the one-column Young symmetrizer in a sense explained below.Let Ch(R-mod-R) denote the category of complexes of finitely generated, graded (R, R)-bimodules. Our first main theorem says that P ∨ 1 n is related to a certain braid group action on Ch(R-mod-R). Associated to each n-strand braid,denote the Rouquier complex associated to the (positive) full twist braid. Then {FT ⊗k } ∞ k=0 , after the appropriate grading shifts, can be made into a direct system with homotopy colimit P ∨ 1 n . Remark 1.3. Dually, one may consider a complex P 1 n which is defined to be a projective resolution P 1 n → R viewed as a graded R ⊗ R W R-module. Analogously, if we let FT −1 denote the Rouquier complex of the negative full twist braid then {(FT −1 ) ⊗k } ∞ k=0 can made into an inverse system with homotopy limit P 1 n , after the appropriate grading shifts. We will prove in §2.6 that the duality functor (·) ∨ : Ch(R-mod-R) → Ch(R-mod-R) sends P 1 n to P ∨ 1 n . Our notation prefers complexes which are bounded above, in order to be consistent with [15]. However, in this paper the complex P ∨ 1 n is often nicer than P 1 n , since P ∨ 1 n is an algebra in the homotopy category K + (R-mod-R), while P 1 n is a coalgebra. This is discussed in Remark 1.13. The algebra structure makes writing our following main results easier. See Example 4.13 for a sample computation in the non-dual, n = 2 case.Let us recall the construction of Khovanov and Rozansky's triply-graded link homology given in [17]. Hochschild cohomology of bimodules defines a functor HHH from Ch(R-mod-R) to the category of triply-graded vector spaces, and HHH(F (β)) is a well-defined invariant of the oriented linkβ, up to isomorphism and overall shift of triply-graded vector spaces. This invariant is called Khovanov-Rozansky homology, hereafter referred to as KR homology.Remark 1.4. Actually, in [17] the results are stated in terms of Hochschild homology. In this paper, as in [15], we prefer Hochschild cohomology to Hochschild homology; there is not much difference since for polynomial rings the two are isomorphic up to regrading. Our convention will ensure that HHH applied to our projector is a graded commutative algebra, rather than an algebra up to regrad...

Categorified Young symmetrizers and stable homology of torus links

“…The regrading involved in the transposition symmetry isq ↔t andā →ā. Indeed, the computations in [AH15] involve even variables of degreē tq 1−k and odd variables of degreeāq 1−k , as predicted.…”

“…One aspect of the conjectures in [GNR16] is that, up to regrading, there should be an isomorphism between the DHom(P T , X ⊗ P T ) and DHom(P T * , X ⊗ P T * ) up to regrading, where T is a standard tableau on n-boxes, T * is the transposed tableau, and X = σ n−1 · · · σ 2 σ 1 is the positive braid lift of a Coxeter element of S n . The results of this paper and [AH15] demonstrate this duality in the case of the one-column and one-row partitions.…”

“…in a special case. The computations in [AH15] prove the conjecture in the case of the onecolumn projector. One aspect of the conjectures in [GNR16] is that, up to regrading, there should be an isomorphism between the DHom(P T , X ⊗ P T ) and DHom(P T * , X ⊗ P T * ) up to regrading, where T is a standard tableau on n-boxes, T * is the transposed tableau, and X = σ n−1 · · · σ 2 σ 1 is the positive braid lift of a Coxeter element of S n .…”

“…Both projectors are homotopy colimits of directed systems R → FT n → FT ⊗2 n → · · · (degree shifts omitted), but using different maps. One may think of P n as the "head" and P 1 n as the "tail;" we won't attempt to make these terms precise, but instead direct the reader to [AH15] for an explanation of the relationship between the two projectors. There are many important differences between the two stories.…”

On the computation of torus link homology

“…The Rouquier complex is normalized so that if is a positive braid, then there is a chain map which is the inclusion of the degree chain bimodule, whereas if is a negative braid, there is a chain map which is the projection onto the degree bimodule. Note, in [AH15] and [Hog18], the shifts and would have been denoted and , respectively.…”