Richardson Varieties in the Grassmannian
Richardson Varieties in the Grassmannian
复制标题
格拉斯曼的理查森品种
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
V. Lakshmibai
中科院分区：
文献类型：
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作者：
V. Kreiman;V. Lakshmibai
Richardson簇$X_w^v$被定义为Schubert簇$X_w$和相反的Schubert簇$X^v$的交集。对于格拉斯曼环X_w^v$，我们得到了X_w^v$的齐次坐标环的一个标准单项基.我们首先利用这个基证明了H^i（X_w^v，L^m），i > 0，m geq 0的消失性，其中L是Grassmannian的Picard群的样本生成元对X_w^v的限制，然后确定切空间的基和X_w^v在任意T$-不动点e_（？） 最后给出了X_w^v$在任意T$-不动点e_v的重数的递推公式 $.利用递归公式，我们证明了$X_w^v$在$e_w处的重数 $是$X_w$在$e_处的重数的乘积 $和$X^v$在$e_处的重数 $.这一结果使我们能够推广Rosenthal-Zelevinsky行列式公式的多重性在$T$-不动点的舒伯特品种的情况下，理查森品种。
The Richardson variety $X_w^v$ is defined to be the intersection of the Schubert variety $X_w$ and the opposite Schubert variety $X^v$. For $X_w^v$ in the Grassmannian, we obtain a standard monomial basis for the homogeneous coordinate ring of $X_w^v$. We use this basis first to prove the vanishing of $H^i(X_w^v,L^m)$, $i > 0 $, $m geq 0$, where $L$ is the restriction to $X_w^v$ of the ample generator of the Picard group of the Grassmannian; then to determine a basis for the tangent space and a criterion for smoothness for $X_w^v$ at any $T$-fixed point $e_ $; and finally to derive a recursive formula for the multiplicity of $X_w^v$ at any $T$-fixed point $e_ $. Using the recursive formula, we show that the multiplicity of $X_w^v$ at $e_ $ is the product of the multiplicity of $X_w$ at $e_ $ and the multiplicity of $X^v$ at $e_ $. This result allows us to generalize the Rosenthal-Zelevinsky determinantal formula for multiplicities at $T$-fixed points of Schubert varieties to the case of Richardson varieties.