Coordinate rings and birational charts

Spherical birational sheets in reductive groups

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when $G$ is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes $\mathcal{O}_1$, $\mathcal{O}_2$ of $G$ lie in the same birational sheet, up to a shift by a central element of $G$, if and only if the coordinate rings of $\mathcal{O}_1$ and $\mathcal{O}_2$ are isomorphic as $G$-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of $G$.

分裂对称空间的雅可比下降图与对数商坐标

Jacobi descent charts and logarithmic quotient coordinates for split symmetric spaces

本文针对等秩分裂对称对，构造了趋近幂零纤维的正则族的有限下降图，利用库默理论分类相关覆盖，得到嘉当提升的显式形式，证明了商密度的乘法哈尔测度性质。

通过簇代数中的变异组合学得到双有理 Weyl 群作用与 q-Painlevé 方程

Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras

本文利用簇代数中箭图变异的图组合方法，系统构造了 Weyl 群的无减法双有理表示，并研究其辛结构与 Darboux 坐标，为 q-Painlevé 方程及其高阶推广提供了广泛的群作用例子。

Bott-Samelson varieties and Poisson Ore extensions

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\bf u}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $π_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $π_{\rm st}$ on $G$. We explicitly express $π_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $π_n$ to each affine coordinate c…

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups $K=U(n), SO(n)$. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniques of Poisson-Lie theory and the geometric crystals of …

辛群oid上的理查德·韦伊群作用与簇代数

Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras

本文研究了辛群oid上的理查德·韦伊群作用，通过簇变换关联于quiver中的某些环，证明了韦伊群不变子代数是矩阵条目生成代数的有限中心扩张，并展示了该作用在经典情况下的嵌入图像与韦伊群不变子代数的商代数的泊松同构性。