Schubert calculus in equivariant K-theory
| Project/Area Number |
15K04832
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Okayama University of Science |
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Principal Investigator |
Ikeda Takeshi 岡山理科大学, 理学部, 教授 (40309539)
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| Co-Investigator(Kenkyū-buntansha) |
成瀬 弘 山梨大学, 大学院総合研究部, 教授 (20172596)
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| Research Collaborator |
Nakasuji Maki
Matsumura Tomoo
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | K理論 / シューベルト類 / グラスマン多様体 / シューベルト多様体 / アフィン・グラスマン / パッフィアン / ピエリ規則 / K-理論 / シューベルト・カルキュラス / 退化跡 / リトルウッド・リチャードソン規則 |
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| 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.
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| 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．
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Report
(4 results)
Research Products
(37 results)
