Ben Elias, Geordie Williamson
We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any 2-sided cell in a Coxeter group is rigid and pivotal.
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Singular Soergel bimodules
本文定义并研究了作为 Soergel 双模自然推广的奇异 Soergel 双模范畴，分类了其不可分解对象，证明了其 2-范畴的分裂 Grothendieck 群同构于 Schur 代数体，并揭示了特征零下原猜想对奇异情形的推论。
Almost finitary birepresentation theory and applications to affine Soergel bimodules
本文发展了适用于无限Coxeter群Soergel双模的有限双表示理论泛化，建立了简单双表示分类的简化过程，并详细讨论了扩展仿射类型A中的Soergel双模情况。
We show that the endomorphism ring of the projective generator in the category of Soergel modules (for dihedral groups) is Koszul self-dual.
We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develop graphical calculi in the 2-variable case for the odd Soergel category and the related singular Soergel 2-category. We describe the odd analogue of the Rouquier complexes and establish their invertibility in the homotopy category. For three variable…
We consider a generalisation of the specialisation functor on Soergel bimodules and show that this generalised version still takes Soergel bimodules to Soergel modules.
Evaluation birepresentations of affine type A Soergel bimodules
本文定义仿射A型Soergel双模到有限A型的赋值函子，范畴化对应Hecke代数的赋值同态，可拉回得到三角双表示，并证明胞腔双表示对应赋值双表示有有限覆盖，以次正则顶点情形为例说明。
A foamy approach to Soergel bimodules
本文通过建立泡沫2范畴与类型A奇异Soergel双模的2等价，探索双模结构的深层联系。
We produce an explicit recursive formula which computes the idempotent projecting to any indecomposable Soergel bimodule for a universal Coxeter system. This gives the exact set of primes for which the positive characteristic analogue of Soergel's conjecture holds. Along the way, we introduce the multicolored Temperley-Lieb algebra. An appendix by Ben Webster gives a precise condition on the base ring for a Jones-Wenzl projector to exist.
In this note, we compute the split Grothendieck ring of a generalized category of Soergel bimodules of type $A_2$, where we take one generator for each reflection. We give a presentation by generators and relations of it and a parametrization of the indecomposable objects in the category, by realizing them as rings of regular functions on certain unions of graphs of group elements on a reflection faithful representation.
The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke …
We produce Jucys-Murphy elements for the diagrammatical category of Soergel bimodules associated with general Coxeter groups, and use them to diagonalize the bilinear form on the cell modules. This gives rise to an expression for the determinant of the forms and Jantzen type sum formulas.
Higher Idempotent Completion for Soergel Bimodules
本研究将高等幂等完备化应用于纽结同调与高等表示论相关的高等范畴，证明奇异Soergel双模可由部分2-范畴幂等完备化复原，构建了A型下的半严格幺半2-范畴并给出泡沫理论的高等范畴分支规则，得到Rose-Wedrich分解定理的完全局部版本。
On one-sided singular Soergel bimodules
本文建立了一种作为 Williamson 理论部分推广的奇异 Soergel 双模理论，并采用了作者开发的 Soergel 双模表述。
We prove that every simple transitive $2$-representation of the fiat $2$-category of Soergel bimodules (over the coinvariant algebra) in type $B_2$ is equivalent to a cell $2$-representation. We also describe some general properties of the $2$-category of Soergel bimodules for arbitrary finite Dihedral groups.
Singular Soergel bimodules for realizations
本文将Williamson定义的附加于Coxeter群反射忠实表示的奇异Soergel双模范畴推广至更一般的Coxeter群实现。
On the functor relating Harish-Chandra bimodules and Soergel bimodules
本文重新考察 Soergel 函子，将其与 Losev 的限制函子联系起来，计算相关双模的像并给出函子性质的新证明，同时把整块结果推广到一般块。
Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather elementary, some of their crucial properties were known until recently only in the case of crystallographic Coxeter groups, where these bimodules can be interpreted in terms of equivariant cohomology of Schubert varieties. In recent work Elias and Williamson have proved these properties in full generality by showing that these bimodules poss…
We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.
We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.
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 …
Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we …
We generalize the modular Koszul duality of Achar-Riche to the setting of Soergel bimodules associated to any finite Coxeter system. The key new tools are a functorial monodromy action and wall-crossing functors in the mixed modular derived category. In characteristic 0, this duality together with Soergel's conjecture (proved by Elias-Williamson) imply that our Soergel-theoretic graded category $\mathcal{O}$ is Koszul self-dual, generalizing the result of Beilinson-Ginzburg-Soergel.
We prove the local hard Lefschetz theorem and local Hodge-Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.
Induction for extended affine type A Soergel bimodules: first steps
本文初步探讨了扩展有理类型A Hecke代数有限维表示的Zelevinsky张量积的范畴化。
For a Coxeter system and a representation $V$ of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when $V$ is reflection faithful. Elias and Williamson defined another category even when $V$ is not reflection faithful and they proved that this category is equivalent to the category of Soergel bimodules when $V$ is reflection faithful. Moreover they proved the categorification theorem for their category with less assumptions on $V$. In this paper, we give a "bimodule theoretic…
We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two one-sided twisted complexes constructed from Rickard complexes of singular Soergel bimodules associated to braided webs. Along the way, we prove a conjecture of Beliakova--Habiro relating the colored 2-strand full twist complex with the categorical ribbon element for quantum $\mathfrak{sl}_2$.
Braiding on complex oriented Soergel bimodules
本文在稳定同伦论中定义谱值 U(n) Soergel 双模的高阶范畴，并在复定向条件下为其相关局部稳定范畴构造辫结构，同时建立标准分裂的谱类比。
In this paper we continue the study of the category of modular Harish-Chandra bimodules initiated by Bezrukavnikov and Riche and also study the modular version of the BGG category $\mathcal{O}$. We prove a version of the Bezrukavnikov-Mirkovic-Rumynin localization theorem for the Harish-Chandra bimodules and for the category $\mathcal{O}$. We also relate the category of Harish-Chandra bimodules to the affine Hecke category building on the prior work of Bezrukavnikov and Riche.
The Two-Color Ext Soergel Calculus
该研究在特征0下计算二面体群相关Soergel双模的Ext群，得到其Hochschild上同调的图解基与对应幺半范畴的图解表示，还应用于链环同调计算并给出二面体情形Soergel Hom公式的t-模拟。
The monoidal category of Soergel bimodules is an incarnation of the Hecke category, a fundamental object in representation theory. We present this category by generators and relations, using the language of planar diagrammatics. We show that Libedinsky's light leaves give a basis for morphism spaces and give a new proof of Soergel's classification of the indecomposable Soergel bimodules.
We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl(2). When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soe…
We give a gentle introduction to the concept of folding. That is, we provide an elementary discussion of equivariant categories, their weighted Grothendieck groups, and the technical aspects of computing with them. We then perform the computations required to confirm that quasi-split Hecke algebras with unequal parameters are categorified by equivariant Soergel bimodules, in almost every case.
This paper is the first of a series of introductory papers on the fascinating world of Soergel bimodules. It is combinatorial in nature and should be accessible to a broad audience. The objective of this paper is to help the reader feel comfortable calculating with Soergel bimodules and to explain some of the important open problems in the field. The motivations, history and relations to other fields will be developed in subsequent papers of this series.
A geometric model for Hochschild homology of Soergel bimodules
在Khovanov和Rozansky的三阶化链同同调论的计算中，一个重要的步骤是确定SL（n）的Soergel双模的霍赫希尔德同调.我们对任何单群G给出了这种霍赫希尔德同调的几何模型，即G中B × B-轨道闭包的等变交同调.我们表明，在A型这些轨道闭包是等变形式的共轭T-行动。我们利用这一事实表明，在相应的轨道闭包是光滑的情况下，这个霍赫希尔德同调是一个外代数上的多项式环的发电机，其程度是明确确定的几何轨道闭包，并描述其希尔伯特级数，证明了一个猜想的雅各布拉斯穆森。
A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan-Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the exist…
We determine for which Coxeter types the associated small quotient of the $2$-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive $2$-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive $2$-representations. For the small quotients of general Coxeter types, we give a description for the cell $2$-representations.
Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types
该研究考察有限 Coxeter 类型 Soergel 双模的简单传递 $2$-表示，证明其等价类有限，并在除 $H_3$、$H_4$ 外的所有类型中完成分类。
For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={D_w}_{w\in W} such that each D_w contains as a direct summand (or is equal to) the indecomposable Soergel bimodule B_w. When decategorified, we prove that D gives rise to a set {d_w}_{w\in W} that is actually a basis of the Hecke algebra. This basis is close to the Kazhdan-Lusztig basis and satisfies a ``positivity condition''.
Soergel bimodules and Kazhdan-Lusztig polynomials
本文以凝练纲要介绍 Soergel 双模及其在 Kazhdan–Lusztig 理论中的应用，重点说明其如何为 Kazhdan–Lusztig 猜想提供替代证明。
Diagramatics of $A_1 \times A_1$ Folded Soergel Bimodules
本文给出 $A_1 \times A_1$ 型 Soergel 双模在 $\mathbb{Z}/2$ 作用下的等变化范畴的同痕表示，为研究等变 Soergel 双模范畴奠定基础。
K-theory Soergel Bimodules
本文提出 K-理论 Soergel 双模概念，证明其态射可由 Bott-Samelson 簇间的等变 K-理论对应几何描述，并引入分层等变 K-动机形式体系，实现旗流形上 Bruhat-分层等变 K-动机的链复形刻画，提出 Koszul 对偶等猜想。
We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in case the biclosed set is an inversion set of an e…
This is the author's diploma thesis. We describe a simplification in the construction of Khovanov-Rozansky's categorification of quantum sl(n) link homology using the theory of maximal Cohen-Macaulay modules over hypersurface singularities and the combinatorics of Soergel bimodules. More precisely, we show that the matrix factorizations associated to basic MOY-graphs equal Cohen-Macaulay approximations of certain Soergel bimodules, and prove that taking Cohen-Macaulay approximation commutes with tensor products as long as the MOY-graph under consideration does not possess oriented cycles. It f…
Sur la catégorie des bimodules de Soergel
Coxeter系统（W，S）的Soergel范畴B是W作用于其上的多项式代数上的双模范畴。这是Hecke代数（W，S）的一个分类。本文给出了B中态射空间的一种组合刻画。作为推论，我们给出了O_0-proj中态射的类似描述，其中O_0是BGG范畴O的主块. - Soergel的Coxeter（W，S）系统的B类是一个关于W agit的多项式代数的双模类。这是Hecke de（W，S）对阿尔及利亚的分类。在这篇文章中，我们给出了B语言中形态空间的组合描述。因此，O_0是BGG中O类的主要成分。
In this paper we express certain multiplicities in modular representation-theoretic categories of type A in terms of affine p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal with include the categories of rational representations of GL(n), representations of the quantum group for gl(n), and representations of (degenerate) cyclotomic Hecke and Schur algebras, where the base field is an algebraically closed field of arbitrary prime characteristic. In order to approach this problem we define Soergel-theoretic versions of parabolic categories O in characteristic p. We …
We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation.
In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-r…
Cyclotomic nil-Brauer and Singular Soergel bimodules of type D
本文提出了一种新的单则范畴，通过构造从 cyclotomic nil-Brauer 分类到类型 D 的奇异 Soergel 双模的单则函子，并探讨了其在 Grothendieck 环上的同构性。
A Hom formula for Soergel modules
本文研究任意 Coxeter 群的 Soergel 模，证明无限情形下其同态多于 Soergel 双模态射，否定 Soergel 的一个问题；并利用结构代数上的模结构使态射维数与 Hecke 配对一致，构造特殊子模以推广 Bruhat 区间形状结果。
The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.
Braiding on type A Soergel bimodules: semistrictness and naturality
本文在具体 dg-模型中显式构造了 A 型 Soergel 双模链复形范畴上的编织结构，建立其半严格幺半 2-范畴框架，并给出编织自然性的高阶同伦数据。
Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups
本文为仿射 Weyl 群构造从图解 Hecke 范畴到矩阵范畴的函子，范畴化递归表示，从而将图解 Soergel 双模约化为按因子 p 更小的双模，并给出 p-典范基及不可分解倾斜模特征标的新下界。
We generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules by taking minimal complexes of the restriction of Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are delta-split, they satisfy a vanishing formula and, when Soergel's conjecture holds they are perverse. As an application, we use singular Rouquier complexes to establish Hodge theory for singular Soergel bimodules.
Presentation of right-angled Soergel categories by generators and relations
Soergel双模范畴B是Coxeter系统（W，S）的Hecke代数的范畴化。当W是直角Coxeter群时，我们发现B（作为张量范畴）由生成元和关系表示。
Serre Duality and the Whitehead Link
本文证明了二面体Soergel双模同伦范畴中全扭转作为Serre函子，并利用部分迹函子的可表示性计算了Whitehead链环的三重分次链环同调。
A braided monoidal $(\infty,2)$-category of Soergel bimodules
本文首次将 Hecke 辫子幺半范畴范畴化为 $\mathbb{E}_2$-幺半 $(\infty,2)$-范畴，基于 Soergel 双模链复形，旨在控制同伦相干链环同调及 4/5 维 TQFT，并发展了分解系统工具。
We give an interpretation of sl_n webs as morphisms between certain singular Soergel bimodules. We explain how this is a combinatorial, algebraic version of the geometric Satake equivalence (in type A). We then q-deform the construction, giving an equivalence between representations of U_q(sl_n) and certain singular Soergel bimodules for a q-deformed Cartan matrix. In this paper, we discuss the general case but prove only the case n=2,3. In the sequel we will prove the case n >= 4.
We realize the Temperley-Lieb algebra by analogues of Soergel bimodules. The key point is that the monoidal structure is not given by a usual tensor product but by a slightly more complicated operation.
In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.
We categorify the Hecke algebra with parameters 1 and v using a variation of the category of Soergel bimodules.
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