Combinatorial K-theory

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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.