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 $G=\operatorname{SL}(n)$ . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups $G$, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of $G$.
We give a description of the extremal rays of the eigencones for arbitrary semisimple groups $G$ by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face, and the remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize work of [Bel17] which handled the case of $G=\operatorname{SL}(n)$.
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 $Γ_n(s)$ controlling this problem have been characterized in terms of classical Schubert calculus by the work of several authors. We determine extremal rays of $Γ_n(s)$ (which are never regular faces) by relating them to the geometry of flag varieties: The extremal rays either arise from "modular intersection loci", or by "induction" from extremal rays of smaller groups. Explicit formulas are given for both the extr…
We examine the extremal rays of the cone of dominant weights $(μ, \widehatμ)$ for groups $G\subseteq \widehat G$ for which there exists $N \gg0$ such that $$ \left(V(Nμ)\otimes V(N\widehat μ)\right)^G\ne (0). $$ We exhibit formulas for a class of rays ("type I") on any regular face of the cone. These rays are identified thanks to a generalization of Fulton's conjecture, which we prove along the way. We verify that the remaining rays ("type II") on the face are the images of extremal rays for a smaller cone under a certain map, whose formula is given. A procedure is given for finding the rays o…
We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.
设X是伪指数为i_X且Picard数至少为2的Fano流形，R是X的具有例外轨迹Exc（R）的极值射线.本文证明了一个不等式，它用i_X和Exc（R）的维数来限定R的长度，并讨论了边界情形.特别地，我们对伪指数为i_X的Fano流形X进行分类，该流形X是沿着光滑子流形T将一个光滑簇Y爆破得到的，使得dim T < i_X。
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.
This work is concerned with the study of the extreme rays of the convex cone of $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). We characterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\cal C}_3;$ when the determinant is a perfect square, then the form is either an ex…
We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number one. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which th…
We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.
