论文标题
僵硬的复曲矩阵舒伯特品种
Rigid toric matrix Schubert varieties
论文作者
论文摘要
对于给定的置换$π\在s_n $中，富尔顿证明了矩阵schubert $ \ overline {x_π} \ congy_π\ times \ times \ times \ mathbb {c}^q $可以通过在Rothe图中编码的某些等级条件来定义。在$y_π：= \ text {tv}（σ_π）$的情况下，$是折叠的（相对于$（\ mathbb {c}^*）^{2n-1} $ action），我们可以证明它可以描述为bipartite Graph Graph Graph $ g^π$ g^π$的优势。我们在$ g^π$的子图中明确表征了相关的所谓边缘锥体的较低面孔，并为$y_π$的一阶变形提供了组合研究。我们证明$y_π$在且仅当$σ_π$的三维面都是简单的时才刚性的。此外，我们根据$π$的Rothe图来重新制定此结果。
For a given permutation $π\in S_N$, Fulton proves that the matrix Schubert variety $\overline{X_π} \cong Y_π \times \mathbb{C}^q$ can be defined via certain rank conditions encoded in the Rothe diagram of $π$. In the case where $Y_π:=\text{TV}(σ_π)$ is toric (with respect to a $(\mathbb{C}^*)^{2N-1}$ action), we show that it can be described as an edge ideal of a bipartite graph $G^π$. We characterize the lower dimensional faces of the associated so-called edge cone $σ_π$ explicitly in terms of subgraphs of $G^π$ and present a combinatorial study for the first order deformations of $Y_π$. We prove that $Y_π$ is rigid if and only if the three-dimensional faces of $σ_π$ are all simplicial. Moreover, we reformulate this result in terms of Rothe diagram of $π$.
