Certain Algebraic Varieties Related to Finite and Affine Flag Varieties

旗簇是代数几何中一类重要的簇。舒伯特变种是变种内部的一类重要的亚变种。主要研究者与Musili和Seshadri合作开发了一个“标准单体理论”（以下简称SMT），用于旗品种和舒伯特品种。这一理论导致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，一旦有一个很好的SMT的代数品种，许多信息可以推断品种（使用SMT），例如，SMT抛出光的退化品种。一个变种的退化反过来又促进了对该变种几何方面的理解。这种技术最近被用于计算机科学中的复杂性理论领域，特别是，在“P不等于NP-猜想”的背景下。还有另一类有趣而重要的代数簇与Flag簇有关，即半单代数群在其李代数中幂零元簇上的伴随作用的轨道闭包类。虽然有很多有趣的代数簇-行列式簇，梯形行列式簇，梯形簇-以自然的方式与舒伯特簇中的某些开子集相识别，但上述轨道闭包（以及更一般地，来自循环箭图的轨道闭包）以自然的方式与仿射舒伯特簇中的某些开子集相识别，即，Kac-Moody代数的广义旗簇中的Schubert簇。本提案中考虑的研究项目旨在通过上述与舒伯特品种的关系为上述轨道闭合开发一个SMT;它还旨在为其他感兴趣的品种类别开发一个SMT-大舒伯特品种，球形品种等，并明确地确定了旗簇的等变Grothendieck环和等变上同调环的乘法结构。现代代数几何（发展于世纪后半期）已经证明了自己（毫无疑问）是不可或缺的各种学科内的数学以及在其他领域以外的数学：例如：拓扑学，表示论，组合学（数学内）。物理学中的现代量子理论（特别是量子共形场论）。计算机科学中的机器人学，复杂性理论，计算机视觉。这个建议是在交换代数的十字路口&，代数几何，组合数学&表示论。在这个建议中研究的品种形成了代数几何中一类重要的品种;例如，舒伯特品种理论（在有限域上）与编码理论密切相关。 主要研究者认为，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.