On syzygies of highest weight orbits

射影空间的线性合系

Linear syzygies of projective space

研究线性合系q=1情形下Veronese簇合系的Ein-Lazarsfeld猜想，证明在最高和次高阶线性合系上与猜想一致的消失性命题。

A duality theorem for syzygies of Veronese ideals

We prove a duality theorem for simplicial complexes arising from a combinatorial construction we define, which applies to the squarefree monomial complexes for Veronese ideals of projective spaces and weighted projective spaces. Our theorem yields a formula for the multigraded Betti numbers of these Veronese ideals in terms of the reduced homology groups of these complexes which is dual to one given by Bruns and Herzog. We apply this formula in several ways, including by giving an algorithm for finding the highest syzygy of such a Veronese ideal.

Equations determining the orbit of the highest weight vector in the adjoint representation

We explicitly construct a set of quadratic equations defining the highest weight vector orbit for adjoint representations of Chevalley groups of types D_l, E_6, E_7, and E_8. The combinatorics of these equations is related to the combinatorics of embeddings of the root system of type A_3. We believe that the constructed equations provide a prominent framework for calculations with exceptional groups in adjoint representations, which is particularly interesting for groups of type E_8.

闭轨道的密切簇的二次包络的不可约性

Irreducibility of quadratic hulls of osculating varieties of closed orbits

该论文研究连通复半单代数群的闭最高权轨道的p-密切簇的二次包络不可约性，证明当d>4p时二次包络与密切簇重合，确定其不可约性及集合论刻画。

Some amazing properties of spherical nilpotent orbits

Let $\g$ be simple Lie algebra. We give a conceptual proof for the fact that the nilpotent orbits of height 3 are spherical. It is shown that if the highest root of $\g$ is fundamental, then $\g$ has a specific nilpotent orbit of height 3. This orbit satisfies several interesting relations. Moreover, it can be used for an intrinsic construction of a $G_2$ grading in $\g$. We also discuss an approach to describing the algebra of covariants on a nilpotent orbit. For the nilpotent orbits of height 2, it is shown that the algebra of regular functions is a free module over the algebra of covariants…

Degenerate $SL_n$: representations and flag varieties

The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded representations of the classical group. Following the classical construction of the flag varieties, we consider the closures of the orbits of the abelian unipotent subgroup in the projectivizations of the representations. We show that the degenerate flag varieties $\Fl^a_n$ and their desingularizations $R_n$ can be obtained via this construction. We prove that …