Richardson Varieties in the Grassmannian
| dc.creator | Kreiman, Victor | |
| dc.creator | Lakshmibai, V. | |
| dc.date | 2002-03-27 | |
| dc.date | 2002-04-10 | |
| dc.date.accessioned | 2026-07-07T04:47:19Z | |
| dc.date.available | 2026-07-07T04:47:19Z | |
| dc.description | 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 $e_\t$; and finally to derive a recursive formula for the multiplicity of $X_w^v$ at any $T$-fixed point $e_\t$. Using the recursive formula, we show that the multiplicity of $X_w^v$ at $e_\t$ is the product of the multiplicity of $X_w$ at $e_\t$ and the multiplicity of $X^v$ at $e_\t$. 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. | |
| dc.description | 25 pages. To appear. A reference to Stanley's related work has been added to the introduction | |
| dc.identifier | https://arxiv.org/abs/math/0203278 | |
| dc.identifier | http://arxiv.org/abs/math/0203278 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63665 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14M15 | |
| dc.title | Richardson Varieties in the Grassmannian | |
| dc.type | text |