带补丁的Soergel微积分

The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

We introduce a type $B$ analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type $\tilde{A}_1$. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.Thus we provide strong evidence in favor of the recent Blob vs. Soergel conjecture.

单映射Hecke范畴的Soergel算术

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。

Two-color Soergel calculus and simple transitive 2-representations

In this paper we complete the $\mathrm{ADE}$-like classification of simple transitive $2$-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of $\mathrm{ADE}$ type to give an explicit construction of a graded (non-strict) version of all these $2$-representations. Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large cl…

Thick Soergel calculus in type A

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 …

$\tilde{A}_1$型Soergel calculus与nil-blob代数的分次和公式

Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra

本研究针对$\tilde{A}_1$型Soergel calculus代数，推广其与nil-blob代数的同构至双参数blob代数，结合Jones-Wenzl幂等元计算得到胞腔模双线性形式的对角化，进而证明Jantzen型滤过的分次和公式。

Localized calculus for the Hecke category

We construct a functor from the Hecke category to a groupoid built from the underlying Coxeter group. This fixes a gap in an earlier work of the authors. This functor provides an abstract realization of the localization of the Hecke category at the field of fractions. Knowing explicit formulas for the localization is a key technical tool in software for computations with Soergel bimodules.

Rouquier Complexes are Functorial over Braid Cobordisms

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.

A 型 Soergel 双模上的编织：半严格性与自然性

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。

计算 Hecke 范畴中的双叶系数

Calculating double-leaf coefficients in the Hecke category

针对Abe-Bott-Samelson双模态射，提出计算双叶系数的算法，通过修正Bruhat上三角性实现前向代入求解，并在Sage中实现，用于计算H_3型Zamolodchikov三色关系并简化表达。