Higher secant varieties of the minimal adjoint orbit

Secant varieties and degrees of invariants

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being nonconstructive, the generators and their degrees have remained a subject of interest. In this article we determine certain divisors of the degrees of the generators. Also, for irreducible representations, we provide lower bounds for the degrees, determined by the geometric properties of the unique closed projective $G$-orbit $\mathbb X$, and more specifically it…

Projective aspects of the Geometry of Lagrangian Grassmannians and Spinor varieties

We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Plücker embedding and of Spinor varieties in both their Plücker and Spinor embeddings.

旗簇的割线簇：通过 Schur 极性理论

Secant varieties of flag varieties via Schur apolarity

本文发展 Schur 极性理论以研究旗簇割线簇，引入几何 Schur 平方和 Schur 对偶 Terracini 引理，并确定 $\operatorname{Fl}(1,2;V_n)$ 在 $\mathcal{O}(1,1)$ 嵌入下所有割线簇的维数，仅两个缺陷情形。

幂零轨道及其割线簇

Nilpotent orbits and their secant varieties

本文利用二重作用理论描述单代数群幂零轨道的割线簇仿射锥，计算其维数、复杂度和秩，并揭示其与矩映射闭包的关系。

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a connected subgroup of $G$ with $\mathsf{Lie} G_0=\mathfrak g_0$. We study the $G_0$-equivariant projections $φ:\overline{\mathcal O_{\sf min}}\to \mathfrak g_0$ and $ψ:\overline{\mathcal O_{\sf min}}\to\mathfrak g_1$. It is shown that the properties of $\overline{φ(\mathcal O_{\sf min})}$ and $\overline{ψ(\mathcal O_{\sf min})}$ essentially depend on whether the …

Homogeneous projective varieties with semi-continuous rank function

Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this p…

A link between Quantum Entanglement, Secant varieties and Sphericity

In this paper, we shed light on relations between three concepts studied in representations theory, algebraic geometry and quantum information theory. First - spherical actions of reductive groups on projective spaces. Second - secant varieties of homogeneous projective varieties, and the related notions of rank and border rank. Third - quantum entanglement. Our main result concerns the relation between the problem of the state reconstruction from its reduced one-particle density matrices and the minimal number of separable summands in its decomposition. More precisely, we show that sphericity…

On momentum images of representations and secant varieties

Let $K$ be a connected compact semisimple group and $V_λ$ be an irreducible unitary representation with highest weight $λ$. We study the momentum map $μ:\mathbb P(V_λ)\to\mathfrak k^*$. The intersection $μ(\mathbb P(V_λ))^+=μ(\mathbb P(V_λ))\cap{\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_λ$. We construct an affine rational polyhedral convex cone $Υ_λ$ with vertex $λ$, such that $μ(\mathbb P(V_λ))^+\subsetΥ_λ\cap {\mathfrak t}^+$. We show that equality holds for a class of representations, including those with regular …

Cohomology of hyperplane sections of (co)adjoint varieties

In this paper we study general hyperplane sections of adjoint and coadjoint varieties. We show that these are the only sections of homogeneous varieties such that a maximal torus of the automorphism group of the ambient variety stabilizes them. We then study their geometry, provide formulas for their classical cohomology rings in terms of Schubert classes and compute the quantum Chevalley formula. This allows us to obtain results about the semi-simplicity of the (small) quantum cohomology, analogous to those holding for (co)adjoint varieties.

Quiver varieties and adjoint crystals of level $\ell$ for type $A_n^{(1)}$

Let $B(Λ)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(Λ)$ of the crystal $B(Λ)$ using quiver varieties and the path realization $\mathcal{P}^{\rm ad}(Λ)$ arising from the adjoint crystal $\Bad{\ell}$ of level $\ell$.

On the quantum cohomology of adjoint varieties

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

Classification of complex simple Lie algebras via projective geometry geometry

We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.

Which Schubert Varieties are Hessenberg Varieties?

After proving that every Schubert variety in the full flag variety of a complex reductive group $G$ is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of $G$ increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.

Quiver Varieties and Path Realizations arising from Adjoint Crystals of type $A_n^{(1)}$

Let $B(Λ_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(Λ_0)$ of $B(Λ_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(Λ_0)$ of $B(Λ_0)$ arising from the adjoint crystal $\adjoint$.