Schubert calculus in equivariant K-theory

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04832
• 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(Kenkyū-buntansha): 成瀬 弘 山梨大学, 大学院総合研究部, 教授 (20172596)
• Research Collaborator: Nakasuji Maki ; Matsumura Tomoo
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: K理論 / シューベルト類 / グラスマン多様体

Outline of Final Research Achievements

We obtained determinant and Pfaffian(sum) formulae for the Schubert classes in the equivariant K-theory of classical type A,B, and C Grassmannians (with Hudson, Matsumura, Naruse). Related to the GP functions which are identified with the Schubert classes of maximal orthogonal Grassmannians, we introduced a combinatorial notion called set-valued decomposition tableaux, and gave a conjecture on the structure constant, and gave a proof for special case called Piari case (with Cho, Nakasuji). We formulated K-theoretic Peterson isomorphism and proved it (with Iwao, Maeno). In the equivariant quantum cohomology ring, we proved the factorial P- and Q-funsctions represent the Schubert classes (with Mihalcea, Naruse). For the maximal orthogonal Grassmannian, we proved the Pieri rule in the equivariant cohomology (with Cho). Naruse joint with Kirillov introduced a family of functions that are identified with Schubert classes in the equivariant K-theory of the classical flag variety.

Free Research Field

代数学，組合せ論

Academic Significance and Societal Importance of the Research Achievements

等方グラスマン多様体の種々のコホモロジー理論において，シューベルト類の具体的な記述を与えた．特に同変K理論，同変コホモロジー，量子同変コホモロジーなどである．特に，行列式，パッフィアン公式は数10年来の懸案を解決した．構造定数に関してひとつの予想を立てた．これは組合せ論に新しい概念の導入を含む．その予想に対して，部分的，肯定的解決を与えた．また，K理論における量子・アフィン対応を証明した．