2020 Fiscal Year Final Research Report
Research on topological models for combinatorial Hopf algebras
| Project/Area Number |
18K03303
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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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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Okayama University |
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Principal Investigator |
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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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| Project Period (FY) |
2018-04-01 – 2021-03-31
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| Keywords | トポロジー / 幾何 / 複素コボルディズム / シューベルト・カルキュラス / Schur S-, P, Q-関数 / Hall--Littlewood関数 / Gysin写像 |
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| Outline of Final Research Achievements |
(1) We generalized the Gysin formulas for flag bundles in the ordinary cohomology theory, which are due to Darondeau-Pragacz, to the complex cobordism theory. Then, we introduced the universal quadratic Schur functions, which are a generalization of the quadratic Schur functions introduced by Darondeau-Pragacz, and established some Gysin formulas for them as an application of our Gysin formulas. (2) We introduced a generalization of the ordinary Hall-Littlewood P- and Q-polynomials, which we called the universal factorial Hall-Littlewood P- and Q-functions, and characterized them in terms of our Gysin formulas in complex cobordism. As an application of our Gysin formulas in complex cobordism, we gave generating functions for the universal factorial Hall-Littlewood P- and Q-functions. Using our generating functions, classical determinantal and Pfaffian formulas for Schur S- and Q-polyomials, and their K-theoretic or factorial analogues can be obtained in a simple and unified manner.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
通常のHall－Littlewood多項式はSchur S-多項式とSchur P, Q-多項式を補間する対称多項式であり，表現論や組合せ論において重要な役割を演じるものである。本研究の特色は，これを旗束のGysin写像を通して幾何的に捉えた点にある。幾何的・位相的な観点に立つことで，形式群を利用した幾何的にも意味のある拡張が可能となった。また，Darondeau－Pragaczの公式の複素コボルディズム版を利用することにより，拡張されたHall－Littlewood関数の母関数表示が得られ，Schur S, Q-多項式等の行列式公式やパフィアン公式を統一的な方法により導出することができる。
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