Ben Elias, Mikhail Khovanov
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.
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For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.
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.
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.
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 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.
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…
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。
Soergel calculus for monodromic Hecke categories
本文引入两个2-范畴，分类单映射Hecke代数。通过Abe的Soergel双模理论和Elias-Williamson图示算术扩展，证明两者等价并关联到奇性sheaves的分类。
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 …
Nil graded algebras associated to triangular matrices and their applications to Soergel Calculus
本文引入并研究了一类与Bott-Samelson双模自同态代数的Gelfand-Tsetlin子代数结构紧密相关的代数范畴，并发展了系列技术以获取图解式Soergel范畴中众多Gelfand-Tsetlin子代数的最优表示。
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。
Soergel Calculus with patches
本文将Hecke范畴的图示表示适配到由Rouquier复数标准代表元构成的dg范畴中，借此恢复了辫群关系的范畴化与Rouquier公式等基本结果。
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 …
Soergel bimodule category B is a categorification of the Hecke algebra of a Coxeter system (W,S). We find a presentation of B (as a tensor category) by generators and relations when W is a right-angled Coxeter group.
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。
The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La catégorie B de Soergel d'un système de Coxeter (W,S) est une catégorie de bimodules sur une algèbre de polynômes sur laquelle W agit. C'est une catégorification de l'algèbre de Hecke de (W,S). Dans cet article…
We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.
We initiate the study of Gaitsgory's central sheaves using complexes of Soergel bimodules, in extended affine type A. We conjecture that the complex associated to the standard representation can be flattened to a central complex in finite type A, which is the complex sought after in a conjecture of Gorsky-Negut-Rasmussen.
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。
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.
Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $\mathcal{O}(\mathfrak{gl}_n)$ which show up in categorification of $\mathfrak{gl}(1|1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.
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。
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 …
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。
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.
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.
The category $\bcalNT:$ Isomorphism criteria and applications
本文针对新引入的范畴$\bcalNT$构建同构准则，用于研究图示Soergel范畴的Gelfand-Tsetlin子代数，为其提供结构分析工具。
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.
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。
We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\su…
Lusztig-Vogan categories of equal rank 2
本文研究对应等秩实群的秩2 Lusztig-Vogan范畴，分类其不可分解对象、刻画生成Soergel双模的作用，还提供适用于任意有限秩该范畴的算法，复现了对应Lusztig-Vogan模的W-图。
Affine and cyclotomic Schur categories
利用仿射编织范畴构造仿射Schur范畴和分圆Schur范畴，给出其图式基和双SST基，并证明与退化分圆Schur代数的同构。
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.
We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's "special" bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.
Endoscopy for affine Hecke categories
本文研究仿射单值 Hecke 范畴与内窥群仿射 Hecke 范畴之间的关系，证明其中性块幺半等价，并借助 Soergel 函子及 Bezrukavnikov–Yun 技术推广半单纯复形的等同。
Existence and rotatability of the two-colored Jones-Wenzl projector
该研究以双色量子二项式系数的可逆性与零化性为条件，给出了双色Jones-Wenzl投影算子的存在性与可旋转性判定，进而证明Abe的Soergel双模范畴与图式Hecke范畴完全等价。
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。
Affine and cyclotomic $q$-Schur categories via webs
本文通过构造仿射 q-web 范畴、仿射 q-Schur 范畴及其分圆商范畴并建立图解整基，首次给出仿射与分圆 q-Schur 代数的图解表示。
Some remarks on realizations defining diagrammatic Hecke categories
本文针对定义图示Hecke范畴的实现，证明了确保范畴性态良好的常见假设非必要，还给出非平衡实现旋转标量的非归纳描述与平衡性检验简化准则，补充了图示Hecke范畴相关研究的关键结论。
A diagrammatic approach to categorification of quantum groups I
本文提出一种图解方法，为无环无重边图构造环族，其投射模范畴实现量子群负部分的范畴化。
Planar diagrammatics of self-adjoint functors and recognizable tree series
本文研究自伴函子情形下由嵌套圆图值恢复函子与范畴的逆向问题，揭示平面图式与可识别树级数之间的对应关系。
Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups on the one hand and monoidal 2-categories of Soergel bimodules on the other hand might lead to new interesting 4-dimen…
The Tambara-Yamagami calculus and finite semigroups
本文研究将 Tambara-Yamagami 范畴中的有限群替换为有限半群后，其图解演算的保留与发展，提出并构建了相应的演算体系。
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.
Diagrammatic bases from stratified normalization
本文提出一种基于分层重写系统的规范化方法，用于计算图解范畴的hom基，并应用于$q$-Schur范畴构造新基。
Diagrammatic Categories which arise from Representation Graphs
本文利用群表示图构造图示范畴，并证明在显式判据下该范畴的商范畴等价于由有限多个不可约模的张量积构成的模范畴满子范畴。
We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.
For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
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…
We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.
Affine and cyclotomic webs
本文引入了仿射web范畴及其循环商范畴，并建立了与有限$W$-代数的联系，从而得到了$W$-Schur代数幂等子代数的图示表示。
We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described abo…
Polar Brauer categories, infinitesimal braids and Lie superalgebra representations
本文通过引入无穷小辫子、优惠券和极点推广Brauer范畴，构造到李（超）代数表示范畴的函子，并统一了不变量理论中的经典图解范畴。
Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a \subseteq A_{\bar 0}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{A,a}_I$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T^A_a(n,d)$. We also show there is an asymptotically faithful functor from $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When…
We construct a categorification of the braid groups associated with Coxeter groups inside the homotopy category of Soergel's bimodules. Classical actions of braid groups on triangulated categories should come from an action of this monoidal category. We construct representations of this monoidal category on category O of a complex semi-simple Lie algebra and on constructible sheaves over flag varieties. We also consider general constructions of self-equivalences as reflections around another category.
Endoscopy for metaplectic affine Hecke categories
本文证明了扭曲环路群的仿射 Hecke 范畴等价于 Soergel 双模组合范畴，并由此导出变扭曲群的导出 Satake 等价与 Gaitsgory 量子几何 Langlands 猜想的证明。
Universal Koszul Duality for Kac-Moody Groups
本文通过Soergel理论描述，证明了Kac-Moody旗 manifold上真equivariant K-动机与Langlands对偶的constructible monodromic sheaves之间的单调等价，扩展了有限维旗 manifold上的结果。
Diagrammatics for $F_4$
本文定义图示幺半范畴并构造到 $F_4$ 型例外李代数模范畴的满且本质满射幺半函子，为 $F_4$ 型表示论提供类似经典 Brauer 范畴的图示工具。
Quantum category O vs affine Hecke category
本文研究奇数次单位根处量子范畴O与仿射Hecke范畴的关联，通过范畴形变、构造函子并结合自主发展的Rouquier-Soergel理论，证明仿射范畴O整块与仿射Hecke范畴新t-结构心的最高权范畴等价。
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