Modelling classical types: Algebraic group actions via algebras with symmetries

建模经典类型：通过具有对称性的代数进行代数群作用

• 批准号: 432521517
• 负责人: Dr. Magdalena Boos
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/432521517?language=en
• 关键词: Modelling; classical; types; Algebraic; group

Let G be a simple classical complex Lie group with Lie algebra g. We fix a standard parabolic subgroup P in G and consider its conjugation action on the nilpotent cone N of nilpotent matrices in g of the same size.This action restricts to certain varieties in N, for example to the variety N(x), where x is an integer, of elements n of nilpotency degree x (that is, n^x=0) and to the variety N_P of nilpotent matrices in the Lie algebra of P. We want to understand these group actions.In type A, when without loss of generality G=GL_n (for G=SL_n, the setup is the same), we can translate the group actions to Representation Theory: For each action, we find a bijection between the orbits and certain isomorphism classes of representations of a quiver with relations. Since these bijections are given by associated fibre bundles, orbit closure relations and codimensions are preserved.We have found a similar translation for symplectic and orthogonal G: in these cases there are bijections to certain symmetric isomorphism classes of symmetric representations of symmetric quivers with relations, that is, to isomorphism classes of representations of algebras with self-duality.Concerning the actions mentioned before, depending on the type of G, we want to reach three main goals:I A finiteness criterion which lists all cases in which the described group action only admits a finite number of orbits.II To understand the finite cases in detail, for example while parametrizing the orbits by combinatorial objects, by describing degenerations, by finding singularities in the orbit closures, and by calculation of resolutions of singularities and intersection cohomology.III In the infinite cases, we intent to describe generic normal forms with which we define and understand semi-invariants which generate the parabolic semi-invariant rings. Explicit quotients and equivariant cohomology will be calculated.For type A, there are many known representation-theoretic results which we can use for the examination of the mentioned goals. To understand the actions for the other classical types, we have to extend the symmetric representation theory and the basic focus of this project, thus, lies in this area. The case of the general linear group provides many clues in which way such expansion might be useful and possible; but we also know that new phenomena are to be expected. For example, we proved that the degeneration order in the even orthogonal case is not induced by type A.

设G为具有李代数G的简单经典复李群，在G中固定一个标准抛物子群P，并考虑其对G中相同大小的幂零矩阵的幂零锥N的共轭作用。这种作用局限于N的某些变化，例如幂零度x的元素N（即N ^x=0）的变化N(x)，以及p的李代数中幂零矩阵的变化N_P。我们想要理解这些群作用。在A型中，当不丧失G=GL_n的一般性时（对于G=SL_n，设置是相同的），我们可以将群作用转化为表示理论：对于每个作用，我们找到轨道与具有关系的颤振表示的某些同构类之间的双射。由于这些双射是由相关纤维束给出的，因此保留了轨道闭合关系和协维数。对于辛G和正交G，我们发现了一个类似的平移：在这些情况下，对称颤振的对称表示的某些对称同构类，即具有自对偶性的代数表示的同构类，存在对射。关于前面提到的作用，根据G的类型，我们想要达到三个主要目标：1 .有限准则，它列出了所描述的群作用只允许有限数量的轨道的所有情况。II .详细了解有限情况，例如，通过组合对象对轨道进行参数化，通过描述退化，通过在轨道闭包中寻找奇异点，以及通过计算奇异点和交点上同调的分辨率。在无限情况下，我们打算描述我们用来定义和理解产生抛物型半不变量环的半不变量的一般范式。计算显式商和等变上同调。对于A型，有许多已知的表征理论结果，我们可以用来检查上述目标。为了理解其他经典类型的动作，我们必须扩展对称表示理论，因此，这个项目的基本重点在于这个领域。一般线性群的情况提供了许多线索，说明这种展开可能是有用的和可能的；但我们也知道，新的现象将会出现。例如，我们证明了偶正交情况下的退化序不是由A型引起的。