2022 Fiscal Year Final Research Report
Combinatorics related to representation theory and enumeration of paths
| Project/Area Number |
18K03206
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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 11010:Algebra-related
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| Research Institution | Hokkaido University (2022)
Shinshu University (2018-2021) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Bijective proof / グラフ / Hook Length formula / 強レフシェッツ性 / トグリング群 / directed edge polytope |
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| Outline of Final Research Achievements |
We study an algorithm called Hillman-Grassl algorithm, which is used for a bijective prof of Hook Length formula. We introduced axioms to generalize the algorithm. In our theory, the algorithm is realized as an algorithm to create a path in some graphs.
Moreover we apply our knowledge, obtained by this study, to the other tartgets as follows: 1. We study artinian gorenstein algebras defined by the weighted generating function of matcings. We show the Lefschetz property for the algebras. 2. We also study a quotient algebras monomial ideals. We calculate the determinants of multiplication map in the algebra. We show the Lefschetz property for the algebras. 3. We also study the structure of the toggling group of a path graph. We show the toggling group is the symmetric group on the independent sets of the path graph. 4. We also a polytope defined by a directed graph. We call it the directed edge polytope. We give an characterization for faces and facets of the polytope.
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| Free Research Field |
表現論的組合せ論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では, 組合せ論的構造に着目し, それらがコントールする代数的対象や多面体について研究を行いました. Hook Length formulaに対する研究では, 鍵となるアルゴリズムに対し, 一般化をした上で統一的な解釈を行いました. 代数系や多面体に対する研究では, 多面体の面の特徴付け与えるなどといった結果を得ており, 今後の研究に繋がることが期待できます.
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