相关论文: Vertices of Intersection Polytopes and Rays of Gen…

Let $K$ be a connected compact semisimple group and $V_\lambda$ be an irreducible unitary representation with highest weight $\lambda$. We study the momentum map $\mu:\mathbb P(V_\lambda)\to\mathfrak k^*$. The intersection $\mu(\mathbb…

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of $G=\operatorname{SL}(n)$ .…

We examine the extremal rays of the cone of dominant weights $(\mu, \widehat\mu)$ for groups $G\subseteq \widehat G$ for which there exists $N \gg0$ such that $$ \left(V(N\mu)\otimes V(N\widehat \mu)\right)^G\ne (0). $$ We exhibit formulas…

The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of…

Kostka coefficients appear in the representation theory of the general linear group and enumerate semistandard Young tableaux of fixed shape and content. The $r$-Kostka cone is the real polyhedral cone generated by pairs of partitions with…

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been…

We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian…

Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the…

A large family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is…

For $i=1,\ldots,k$, let $\mathbf{G}_i$ be a connected, simply connected, semisimple algebraic group over some local field $\kappa_i$ of characteristic zero. Let $G_i=\mathbf{G}_i(\kappa_i)$ be the $\kappa_i$-points of $\mathbf{G}_i$ and…

In this paper we use the superpotential for the flag variety $GL_n/B$ and particular coordinate systems that we call ideal coordinates for $\mathbf{i}$, to construct polytopes $\mathcal{P}^{\mathbf{i}}_\lambda$ inside $\mathbb{R}^{R_+}$,…

Consider a complete intersection I of type (d_1,..., d_r) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I^n)}_n obtained by taking the reverse lexicographic generic initial ideals of the…

For a graph $G$ with $p$ vertices the closed convex cone $\mathbb{S}^p_{\succeq0}(G)$ consists of all real positive semidefinite $p\times p$ matrices with zeros in the off-diagonal entries corresponding to nonedges of $G$. The extremal rays…

It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…

Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they…

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-S\"oderberg theory. That is, given a Betti diagram, we determine if it is possible to decompose it into the Betti diagrams of complete…

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the…

Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several…

We determine, in an inductive framework, the vertices of the polytope $P(s,K)$ controlling the conjugacy classes of elements which product to one in the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This extends…

Growing evidence reveals universal hardening on various cosmic ray spectra, e.g. proton, positron, as well as antiproton fraction. Such universality may indicate they have a common origin. In this paper, we argue that these widespread…