Quivers, long exact sequences and Horn type inequalities

A symplectic proof of the Horn inequalities

In this paper, we give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. Our method is a combination of tropical calculus for matrix eigenvalues, combinatorics of planar networks, and estimates for the Liouville volume. As a corollary, we give a tropical description of the Duistermaat-Heckman measure on the Horn polytope.

Total stability functions for type $\mathbb{A}$ quivers

For a quiver $Q$ of Dynkin type $\mathbb{A}_n$, we give a set of $n-1$ inequalities which are necessary and sufficient for a linear stability condition (a.k.a. central charge) $Z\colon K_0(Q) \to \mathbb{C}$ to make all indecomposable representations stable. We furthermore show that these are a minimal set of inequalities defining the space $\mathcal{TS}(Q)$ of total stability conditions, considered as an open subset of $\mathbb{R}^{Q_0} \times (\mathbb{R}_{>0})^{Q_0}$. We then use these inequalities to show that each fiber of the projection of $\mathcal{TS}(Q)$ to $(\mathbb{R}_{>0})^{Q_0}$ is…

关于双曲多项式 的Schur-Horn型不等式

Schur--Horn type inequalities for hyperbolic polynomials

本文为双曲多项式建立了Schur-Horn型不等式，并解决了关于Hadamard型不等式的猜想，通过群对称性原理证明了PSD稳定的lpm多项式Hadamard-Fischer不等式。

The Horn inequalities from a geometric point of view

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 assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated …

U(p,q) 的全纯离散序列表示的张量积与箭图

Tensor product of holomorphic discrete series representations of U(p,q) and quivers

本文利用箭图的表示，为 U(p,q) 李代数中全纯轨道和的轨道不等式提供了新的证明，并探讨了其对全纯离散序列表示张量积分解的意义。

Moment polytopes in real symplectic geometry II : applications to singular value inequalities

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 is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.

On crystal operators in Lusztig's parametrizations and string cone defining inequalities

Let $\mathbf{w}_0$ be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and Kashiwara's (string) parametrizations of the canonical basis associated with $\mathbf{w}_0$. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities def…

The depolarising channel and Horns problem

We investigate the action of the depolarising (qubit) channel on permutation invariant input states. More specifically, we raise the question on which invariant subspaces the output of the depolarising channel, given such special input, is supported. An answer is given for equidistributed states on isotypical subspaces, also called symmetric Werner states. Horns problem and two of the corresponding inequalities are invoked as a method of proof.

Dynkin箭图的全稳定性与Auslander-Reiten理论

Total stability and Auslander-Reiten theory for Dynkin quivers

本文研究Dynkin箭图的全稳定性问题，提出仅需验证Auslander-Reiten箭图边界处的几乎可裂序列满足斜率不等式即可判定全稳定，简化了全稳定稳定性函数的判定条件。

平面网络与可逆矩阵的多重 Horn 问题

Multiple Horn problems for planar networks and invertible matrices

本文研究平面网络多重 Horn 问题，提出迹等式、菱形不等式和四面体等式作为必要条件，并猜想其在网络足够大时也充分；同时证明乘法问题在大缩放参数下 Duistermaat-Heckman 测度集中于八面体递推轨迹附近。

Newell-Littlewood numbers III: eigencones and GIT-semigroups

The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: an eigenvalue interpretation; a minimal list of defining linear inequalities; a description by Extended Horn inequalities, as conjectured in part II of this series; and a factorization of NL-numbers, on the boundary.

四面体Horn问题与U(n) 6j符号的渐近性

The tetrahedral Horn problem and asymptotics of U(n) 6j symbols

本文研究四面体Horn问题，推导特征值的Schur多项式新不等式，证明k次内成立则存在O(ln k/k)误差的近似解，并揭示其与U(n) 6j符号半经典渐近性的关联，证明范数渐近性由厄米矩阵四面体的存在性决定。