相关论文: Complexity of the Zero Set of a Matrix Schubert Id…

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…

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the…

In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points…

Matrix Schubert varieties (Fulton '92) carry natural actions of Levi groups. Their coordinate rings are thereby Levi-representations; what is a combinatorial counting rule for the multiplicities of their irreducibles? When the Levi group is…

Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order…

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution…

Let $G$ be a simple algebraic group of adjoint type of rank $n$ over $\mathbb{C}$. Let $T$ be a maximal torus of $G$, and $B$ be a Borel subgroup of $G$ containing $T$. Let $W=N_{G}(T)/T$ be the Weyl group of $G$. Let…

This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we…

Let $v(n)$ be the largest principal specialization of Schubert polynomials for layered permutations $v(n) := \max_{w \in \mathcal{L}_n} \mathfrak{S}_w(1,\ldots,1)$. Morales, Pak and Panova proved that there is a limit \[\lim_{n \to \infty}…

In an underdetermined system of equations $Ax=y$, where $A$ is an $m\times n$ matrix, only $u$ of the entries of $y$ with $u < m$ are known. Thus $E_jw$, called `measurements', are known for certain $j\in J \subset \{0,1,\ldots,m-1\}$ where…

We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that,…

We show that the principal specialization of the Schubert polynomial at $w$ is bounded below by $1+p_{132}(w)+p_{1432}(w)$ where $p_u(w)$ is the number of occurrences of the pattern $u$ in $w$, strengthening a previous result by A.…

Let $G$ be a complex reductive algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup of $G$ containing $T$, $W$ the Weyl group of $G$ with respect to $T$. Let $w$ be an element of $W$. Denote by $X_w$ the Schubert subvariety of…

Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there…

For an $n\times n$ complex matrix $C$, the $C$-numerical range of a bounded linear operator $T$ acting on a Hilbert space of dimension at least $n$ is the set of complex numbers ${\rm tr}(CX^*TX)$, where $X$ is a partial isometry satisfying…

Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the…

In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order $k$, where $0\leq k\leq n$, and explore their properties. We fully characterize the $3 \times 3$ (strictly) semimonotone matrices of exact order $2$,…

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring…

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the…

The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $\chi_y$-genus admits a generalization for singular complex…