Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space.

中文翻译：

---

派生等价的组合构造

给定归约群的某种线性表示，在 \v{S}penko 和 Van den Bergh 最近的工作中被称为准对称表示，我们构造了其各种几何不变量的相干滑轮的派生类别之间的等价合适的通用稳定性参数的理论 (GIT) 商。这些 GIT 商的变化是比最近关于 GIT 商的派生类别和变化的工作中研究的平衡墙交叉更复杂的墙交叉的例子。我们的构造是算法式的并且非常明确，使我们能够：1) 描述一个倾斜的向量丛，它生成这样一个 GIT 商的派生类别，2) 为 GIT 商的 K 理论在表示方面提供组合基础G的理论，