Combinatorial K-theory

组合K理论

• 批准号: 0070479
• 负责人: Richard Stanley
• 金额: $ 9.2万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070479&HistoricalAwards=false
• 关键词: Combinatorial; theory

The investigator will study the cohomology theory and K-theory of partialflag varieties and quiver varieties. In particular he will look forK-theory parallels of known results in cohomology. A first target is thecohomology and Grothendieck rings of a partial flag variety. Since anypartial flag variety has a cell decomposition into Schubert cells, itscohomology ring has a basis consisting of the cohomology classes ofSchubert varieties. Likewise the Grothendieck ring of algebraic vectorbundles has a basis of structure sheaves of Schubert varieties. Theinvestigator will study the structure constants for these rings withrespect to their bases indexed by Schubert varieties. The main goal is tofind explicit formula for these constants, but also positivity questionsare of interest. In cohomology the structure constants are known to bepositive for geometric reasons, but no combinatorial proof of this fact isknown. In K-theory the investigator has conjectured that the structureconstants have alternating signs, i.e. they are non-negative in evendegrees and non-positive in odd degrees. Finding a proof of this would bevery interesting. An additional goal is to find efficient computeralgorithms for calculating these structure constants. The investigatorwill also try to find a formula for the K-theory class of the structuresheaf of a general quiver variety. Such a formula will generalize aformula for the cohomology class of a quiver variety, which theinvestigator has proved earlier with Fulton. The investigator hopes thatproving such a formula will be of help for constructing an explicitresolution of the structure sheaf of a quiver variety. This wouldgeneralize classical constructions such as the Koszul complex, which is offundamental importance in homological algebra.The development of cohomology theory was motivated in part by the problemof classifying topological spaces. This is important for addressingquestions such as "what is the shape of our universe?". Cohomology theoryis also an important tool for solving problems in enumerative geometry, inwhich one seeks to determine and count all the solutions to a geometricproblem. For example, if the geometric problem is to find lines whichintersect or are tangent to a given collection of fixed geometric figures,then the number of such lines is desired. A very powerful technique forsolving this type of problems is to construct a space consisting of allobjects which could potentially be a solution. Counting the number ofsolutions to a problem is often equivalent to performing a calculation inthe cohomology ring of this space of potential solutions. The objects tobe counted can in many situations be identified with flags of subspaces ina given vector space. Flag varieties, whose points correspond to suchflags of subspaces, are therefore typical candidates to act as the space ofpotential solutions. This makes it important to be able to do efficientcomputations within the cohomology ring of a flag variety, and gives thereason why the structure constants for this ring has been wanted bygeometers and combinatorialists for decades. The K-theory or Grothendieckring of a variety can be seen as a generalization of the cohomology ring.A good understanding of K-theory will therefore give a more completepicture of cohomology theory. At the same time K-theory is important forthe study of vector bundles on a variety and for homological algebra. Thismakes it very natural to try to generalize the known results aboutcohomology theory to K-theory.

研究者将研究部分旗型和颤型的上同论和k -论。特别是，他将研究上同学中已知结果的叉理论平行。第一个目标是部分旗子品种的同源性和格罗滕迪克环。由于任何偏标志品种都有一个细胞分解成舒伯特细胞，因此它的上同环有一个由舒伯特品种的上同类组成的基。同样，代数向量束的Grothendieck环也以Schubert变束的结构束为基础。研究者将研究这些环的结构常数与舒伯特变化索引的碱基有关。主要目标是找到这些常数的显式公式，但也有一些有趣的正性问题。在上同调中，由于几何原因，结构常数已知是正的，但没有已知的这一事实的组合证明。在k理论中，研究者推测结构常数具有交替的符号，即它们在偶数度是非负的，在奇数度是非正的。找到这个的证明将是非常有趣的。另一个目标是找到有效的计算机算法来计算这些结构常数。研究者还将尝试为一般颤振变体的结构束的k理论类找到一个公式。这样的公式将推广一个颤振变种的上同调类的公式，这个公式是研究者早先用Fulton证明过的。研究者希望证明这样一个公式将有助于构造一个颤振品种的结构轴的明确分辨率。这将推广经典结构，如Koszul复合体，这在同调代数中是至关重要的。上同调理论的发展部分是由于拓扑空间的分类问题。这对于解决诸如“我们的宇宙是什么形状的？”这样的问题很重要。上同调理论也是解决枚举几何问题的一个重要工具，在枚举几何中，人们试图确定和计数一个几何问题的所有解。例如，如果几何问题是找到与给定的固定几何图形集合相交或相切的线，那么需要这样的线的数量。解决这类问题的一个非常强大的技术是构建一个由可能成为解决方案的对象组成的空间。计算一个问题的解的个数通常等同于在这个空间的潜在解的上同环中进行计算。在许多情况下，要计数的对象可以用给定向量空间中的子空间标志来标识。标记变体的点对应于子空间的标记，因此是作为潜在解空间的典型候选者。这使得能够在旗子种类的上同环内进行有效的计算变得非常重要，并且给出了为什么几何学家和组合学家几十年来一直想要这个环的结构常数的原因。变种的k理论或Grothendieckring可以看作是上同环的推广。因此，对k理论的良好理解将使上同调理论有一个更完整的图景。同时，k理论对于研究各种向量束和同调代数具有重要意义。这使得尝试将已知的上同调理论的结果推广到k理论是很自然的。