Certain Algebraic Varieties Related to Finite and Affine Flag Varieties

标志变量是代数几何中一类重要的变量。舒伯特品种是旗内品种中一个重要的亚种。首席研究员与Musili和seshadri合作，为旗旗品种及其舒伯特品种开发了“标准单项理论”（以下简称SMT）。这个理论引出了许多重要的几何表示理论的结果。首席研究员最近的研究表明，一旦有一个好的代数品种的SMT，就可以推断出关于该品种的许多信息（使用SMT）；例如，SMT揭示了品种的退化。一个品种的退化反过来又促进了对该品种几何方面的理解。这种技术最近被用于计算机科学的复杂性理论领域，特别是在“P不等于tonp猜想”的背景下。还有另一类有趣而重要的代数变体与Flag变体相关，即半简单代数群对其李代数中幂零元变体的伴随作用的轨道闭包类。虽然有很多有趣的代数变种——行定式变种、梯式决定变种、颤振变种——可以用舒伯特变种中的某些开子集自然地识别，但上述轨道闭包（以及更一般地由循环颤振引起的轨道闭包）可以用仿射舒伯特变种中的某些开子集自然地识别，即与Kac-Moody代数相关的广义标志变种中的舒伯特变种。本提案所考虑的研究项目旨在通过上述与舒伯特变种的关系，为上述轨道闭合开发一种SMT；本文还旨在为其他有趣的品种类型——大Schubert品种、球面品种等开发SMT，以及以显式的方式确定等变Grothendieck环和等变上同调环的乘法结构。现代代数几何（发展于20世纪下半叶）已经证明了自己（毫无疑问）在数学的各个学科以及数学以外的其他领域是不可或缺的：例如：拓扑学，表示理论，组合学（数学内）。物理学中的现代量子理论（特别是量子共形场理论）。机器人，复杂性理论，计算机科学中的计算机视觉。这个提议是交换代数、代数几何、组合学和表示论的交叉点。本文所研究的变种构成了代数几何中一类重要的变种；例如，Schubert变分理论（有限域上）与编码理论密切相关。首席研究员认为，这一建议必将对上述学科产生重大影响。

Flag varieties form an important class of varieties in AlgebraicGeometry. Schubert varieties form an important class of subvarieties insideflag varieties. The principal investigator developed a ``Standard MonomialTheory" (henceforth abbreviated SMT) in collaboration with Musili andSeshadri for flag varieties and their Schubert varieties. This theory hasled to very many important geometric & representation-theoreticconsequences. The principal investigator's recent research shows that oncethere is a good SMT for an algebraic variety, much information could beinferred about the variety (using SMT); for instance, SMT throws light onthe degenerations of the variety. The degenerations of a variety in turnfacilitate the understanding of the geometric aspects of the variety. Thistechnique has been used very recently in the area of Complexity Theory inComputer Science, esp., in the context of the ``P not equal toNP-conjecture". There is yet another interesting and important class ofalgebraic varieties related to Flag varieties, namely, the class of orbitclosures for the Adjoint action of a semi-simple algebraic group on thevariety of nilpotent elements in its Lie algebra. While there are very manyinteresting algebraic varieties - the determinantal varieties, Ladderdeterminantal varieties, quiver varieties - which get identified in anatural way with certain open subsets in Schubert varieties, theabove-mentioned orbit closures (and more generally orbit closures arisingfrom cyclic quivers) get identified in a natural way with certain opensubsets in affine Schubert varieties, i.e., Schubert varieties in thegeneralized flag variety associated to a Kac-Moody algebra. The researchproject considered in this proposal aims at developing a SMT for theabove-mentioned orbit closures via the fore-said relationship with Schubertvarieties ; it also aims at developing a SMT for other interesting classesof varieties - large Schubert varieties, Spherical varieties etc., as wellas determining in an explicit way the multiplicative structure of theequivariant Grothendieck ring and the equivariant cohomology ring of flagvarieties.Modern Algebraic Geometry (developed in the later half of the 20-th century)has proved itself (beyond any doubts) to be indispensable in variousdisciplines within Mathematics as well as in other areas outsideMathematics:Examples: Topology, Representation Theory, Combinatorics (withinMathematics).The modern Quantum Theory (especially Quantum & Conformal field theories) inPhysics.Robotics, Complexity Theory, Computer vision in Computer Science.This proposal is at the cross-roads of Commutative Algebra, AlgebraicGeometry, Combinatorics & Representation-theory. The varieties studied inthis proposal form an important class of varieties in Algebraic Geometry;for example, the theory of Schubert varieties (over finite fields) isclosely linked to Coding theory. The principal investigator believes thatthis proposal is bound to have significant impacts on the above-mentioneddisciplines.