Schubert classes in the equivariant K-theory of flag varieties and related special polynomials

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540032
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Okayama University of Science
• Principal Investigator: IKEDA Takeshi 岡山理科大学, 理学部, 教授 (40309539)
• Co-Investigator(Renkei-kenkyūsha): NARUSE Hiroshi 山梨大学, 教育人間科学部, 教授 (20172596)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 旗多様体 / K理論 / シューベルト類

Outline of Final Research Achievements

We studied the Schubert classes in the equivariant K-theory of generalized flag varieties G/P. First aim is to find good polynomial representatives for the Schubert basis. Second aim is to study the multiplicative structure constants of the Schubert basis by using the obtained polynomials.

We introduced the K-theoretic factorial P- and Q-functions which have several expressions both closed and combinatorial, and represent the Schubert basis of the maximal isotropic Grassmannians. Based on this result, we are able to formulate a conjecture for the structure constants for the maximal orthogonal Grassmannians in K-theory. By using the same underlying idea, we obtained a short proof of Littlewood-Richardson rule in K-theory. We also proved a Pfaffian sum formula for the symplectic Grassmannian in equivariant cohomology, and extended it to equivariant K-theory by using geometric technique. We also obtained a result for the equivariant quantum cohomology of maximal isotropic Grassmannians.

Free Research Field

代数幾何学，組合せ論，トポロジー