Related papers: Matrix Schubert varieties, binomial ideals, and re…
We give a conjectured evaluation of the determinant of a certain matrix $\tilde{D}(n,k)$. The entries of $\tilde{D}(n,k)$ are either 0 or specializations $\mathfrak{S}_w(1,\dots,1)$ of Schubert polynomials. The conjecture implies that the…
We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gr\"obner basis of $I$ with respect to the degree…
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of…
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the…
A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via…
This paper investigates the representation theory of the algebra of partially transposed permutation operators, $\mathcal{A}^d_{p,p}$, which provides a matrix representation for the abstract walled Brauer algebra. This algebra has recently…
To integer programming problems, computational algebraic approaches using Grobner bases or standard pairs via the discreteness of toric ideals have been studied in recent years. Although these approaches have not given improved time…
In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…
It is known that a Markov basis of the binary graph model of a graph $G$ corresponds to a set of binomial generators of cut ideals $I_{\widehat{G}}$ of the suspension $\widehat{G}$ of $G$. In this paper, we give another application of cut…
Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the…
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in…
For a given permutation $\pi \in S_N$, Fulton proves that the matrix Schubert variety $\overline{X_{\pi}} \cong Y_{\pi} \times \mathbb{C}^q$ can be defined via certain rank conditions encoded in the Rothe diagram of $\pi$. In the case where…
In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit…
We construct a Gr\"obner Basis of the relation ideal of a polynomial, give an interpolation formula for the basis elements and explain the connection of the interpolation formula to the Buchberger--M\"oller algorithm. We present a situation…
We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several…
The Specht ideal of shape $\lambda$, where $\lambda$ is a partition, is the ideal generated by all Specht polynomials of shape $\lambda$. Haiman and Woo proved that these ideals are reduced and found their universal Gr\"obner bases. In this…
We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in…
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B…
We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation.
A distinguished algebraic variety in $\mathbb{C}^2$ has been the focus of much research in recent years because of good reasons. This note gives a different perspective. (1) We find a new characterization of an algebraic variety $\mathcal…