Related papers: On the quantum Horn problem

The multiplicative Horn problem is the following question: given three conjugacy classes $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ in a Lie group $G$, do there exist elements…

As shown by P-E Paradan, the set of orbits contained in the sum of two holomorphic orbits in the Lie algebra of U(p,q) is determined by a set of inequalities similar to the Horn inequalities for the sum of conjugacy classes of two Hermitian…

We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_\alpha$, $B\in \mathcal{O}_\beta$ be independent and…

Let $K$ be a connected compact Lie group. The triples $(O_1,\,O_2,\,O_3)$ of adjoint $K$-orbits such that $O_1+O_2+O_3$ contains $0$ are parametrized by a closed convex polyhedral cone called the eigencone of $K$. For $K$ simple of type…

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…

In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which…

This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…

Agnihotri-Woodward-Belkale polytope $\Delta$ (resp. Klyachko cone $K$) is the set of solutions of the multiplicative (resp. additive) Horn's problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian)…

The multiplicative multiple Horn problem is asking to determine possible singular values of the combinations $AB, BC$ and $ABC$ for a triple of invertible matrices $A,B,C$ with given singular values. There are similar problems for…

This is an overview of results from our experiment of merging two seemingly unrelated disciplines - higher algebraic K-theory of rings and the theory of lattice polytopes. The usual K-theory is the ``theory of a unit simplex''. A conjecture…

We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying…

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general…

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…

Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They…

Given a reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$, parabolic subgroups $W_K$'s and a polytope $P$ which is the convex hull of a dominant weight. The quotient $P/W_K$ can be…

We prove a conjecture made by the first author: given an n-body central configuration X_0 in the euclidean space R^{2p}, let Im F be the set of ordered real p-tuples {\nu_1,\nu_2,...,\nu_p} such that {\pm i\nu_1,\pm i\nu_2,...,\pm i\nu_p}…

We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale's geometric method, but…

The condition of nilpotency is studied in the general linear Lie algebra $\mathfrak{gl}_{n}(\mathbb{K})$ and the symplectic Lie algebra $\mathfrak{sp}_{2m}(\mathbb{K})$ over an algebraically closed field of characteristic 0. In particular,…

A symmetry extending the $T^2$-symmetry of the noncommutative torus $T^2_q$ is studied in the category of quantum groups. This extended symmetry is given by the quantum double-torus defined as a compact matrix quantum group consisting of…