AI 中文总结
该研究针对可解代数群作用提出贪心开轨道准则，应用于Dynkin箭图无重表示的Aut_Q(M)作用，简化Lusztig幂零簇刚性问题，为等定向A型提供显式算法。
AI 中文摘要
设G是连通可解代数群，有理作用于有限维向量空间U。利用G-稳定完全旗，我们构造逐次商程序以判定U是否包含开G-轨道；当程序成功时，它会构造出最小支撑基数的开轨道向量，并确定通用稳定子在极大环面商中的像。我们还给出检测开可分轨道的无穷小版本。将该准则应用于Aut_Q(M)对Ext^1_Q(M,M)^*的作用，其中M是 Dynkin 箭图的无重表示；由此，Lusztig幂零簇对应分量的刚性问题可简化为秩检验及活性扩张坐标图的无环条件，所得森林的连通分支决定通用不可分解分解。对于等定向A型，这为一类由关联矩阵编码的多段提供显式算法，包括起点或终点重复的非正则例子。
英文摘要
Let $G$ be a connected solvable algebraic group acting rationally on a finite-dimensional vector space $U$. Using a $G$-stable complete flag, we formulate a successive-quotient procedure that decides whether $U$ contains an open $G$-orbit. When the procedure succeeds, it constructs an open-orbit vector of minimum support cardinality and determines the image of a generic stabilizer in the maximal torus quotient. We also give an infinitesimal version detecting open separable orbits. We apply the criterion to the action of $Aut_Q(M)$ on $Ext^1_Q(M,M)^*$ where $M$ is a multiplicity-free representation of a Dynkin quiver. Rigidity of the corresponding component of Lusztig's nilpotent variety is thereby reduced to a rank test together with an acyclicity condition on a graph of active extension coordinates; the connected components of the resulting forest determine the generic indecomposable decomposition. For equioriented type $A$ this yields an explicit algorithm for a family of multisegments encoded by incidence matrices, including nonregular examples with repeated beginnings or ends.
