We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and, more generally, to Schubert varieties in Grassmannians. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a closed family. We propose a generalization of this to what is called a subclose family. A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined.
We study the combinatorics of Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterizatio…
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
In a paper by Ghorpade and Raghavan, they provide an explicit combinatorial description of the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian, by giving a certain "degree-preserving" bijection between a set of monomials defined by an initial ideal and a "standard monomial basis". We prove here that this bijection is in fact a bounded RSK correspondence.
Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of 4-planes in C^9 with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprim…
We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian a…
We establish a Gysin formula for Schubert bundles and a strong version of the duality theorem in Schubert calculus on Grassmann bundles. We then combine them to compute the fundamental classes of Schubert bundles in Grassmann bundles, which yields a new proof of the Giambelli formula for vector bundles.
We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…
本文讨论[A. Buch '02]和[A.克努森-T Tao '03]。也就是说，我们证明了一个组合规则的环面等变K-理论的格拉斯曼关于基础的舒伯特结构层的结构系数。我们由此推出[H]的代数规则。A-A Yong '13]的相同系数。这两个规则在[D]的意义上都是积极的。安德森Griffeth-E.米勒'11]（而且在更强的形式）。我们的工作是基于组合的基因组tableaux和推广[M.- P. Schutzenberger '77]'s jeu de taquin.
