论文标题
部分可观测时空混沌系统的无模型预测
Gröbner geometry for skew-symmetric matrix Schubert varieties
论文作者
论文摘要
Matrix Schubert品种是在所有$ n \ times n $矩阵上作用的$ b \ times b $ of orbits的关闭，其中$ b $是一组可逆的下三角矩阵。 Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties.他们还表明，相应的初始理想是史丹利 - 赖斯纳（Stanley-Reisner-Reisner）的理想，可壳的简单络合物，并在减少的管道梦中得出了相关的主要分解。 These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications.我们将偏斜的矩阵Schubert品种定义为矩阵舒伯特品种与偏斜 - 对称矩阵的子空间的非空交集。 In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials.正如我们在本文的最后解释的那样，我们的方法与Knutson和Miller的方法不同，可以用来给出一些结果的新证明。
Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article.
