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The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these polynomials. A novel aspect of this alternative to earlier results is that it directly…
A Schubert variety in the complete flag manifold $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has a dense orbit. We give a combinatorial classification of these Schubert varieties.…
We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric…
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them…
We prove a root system uniform, concise combinatorial rule for Schubert calculus of_minuscule_ and_cominuscule_ flag manifolds G/P (the latter are also known as "compact Hermitian symmetric spaces"). We connect this geometry to the poset…
The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this…
Much of modern Schubert calculus is centered on Schubert varieties in the complete flag variety and on their classes in its integral cohomology ring. Under the Borel isomorphism, these classes are represented by distinguished polynomials…
A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If \alpha=(\alpha_1,...,\alpha_d) is an integer…
One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank…
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…
We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key…
The main goal of this paper is to extend two fundamental combinatorial results in Schubert calculus on flag manifolds from equivariant cohomology and $K$-theory to equivariant elliptic cohomology. The foundations of elliptic Schubert…
Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric…
Schubitopes were introduced by Monical, Tokcan and Yong as a specific family of generalized permutohedra. It was proven by Fink, M\'esz\'aros and St.$\,$Dizier that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl…
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of…
We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric…
Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gr\"obner bases in not only algebras with straightening law (ASLs or Hodge…
It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual…
Given an ideal $I$ and a weight vector $w$ which partially orders monomials we can consider the initial ideal $\init_w (I)$ which has the same Hilbert function. A well known construction carries this out via a one-parameter subgroup of a…