Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models
| Project/Area Number |
16540154
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Okayama University |
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Principal Investigator |
HORA Akihito Okayama University, Graduate School of Natural Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (10212200)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Hiro-Fumi Okayama University, Graduate School of Natural Science and Technology, Professor, 大学院自然科学研究科, 教授 (40192794)
MURAI Joshin Okayama University, Graduate School of Humanities and Social Sciences, Assistant, 大学院社会文化科学研究科, 助手 (00294447)
SASAKI Toru Okayama University, Graduate School of Environmental Science, Lecturer, 大学院環境学研究科, 講師 (20260664)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | probability model / symmetric group / asymptotic representation theory / quantum probability / random Young diagrams / character of factor representation / Young graph / scaling limit / 量子確率論 / 表現 / 漸近挙動 / 自由確率 / グラフのスペクトル / ユツィス・マーフィー作用素 / 対称群の表現 / ヤング図形 / 自由確率論 / ラプラシアンのスペクトル / 相互作用フォック空間 |
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| Research Abstract |
The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.
1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.
2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.
3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.
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Report
(4 results)
Research Products
(24 results)
