Combinatorics of Lie type and harmonics analysis on finite groups

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04802
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工学群)
• Principal Investigator: Mizukawa Hiroshi 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工学群), 総合教育学群, 教授 (60531762)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 指標 / 球関数 / 有限等質空間 / 群作用で不変な確率空間

Outline of Final Research Achievements

In this research,　we reached deep understanding of Gelfand pairs of wreath products. Specially, we made great advances in the Ehrenfest's urn models which are an classical physical model of gas diffusion.　Much literature on the subject did not dealt with interactions between urns. But our research constructed interactions arising from finite group actions between the many urns. Moreover we analyzed asymptotic behavior of our models and showed the cut-off phenomenon in many models. Moreover we study the class regular partitions which have deeply connection with the modular representation theory of the symmetric groups. And we get partition identities for multi-component version of class regular partitions.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

２つ壺の中に入った複数のボールが壺の間を移動する，というモデルをエーレンフェストの壺モデルという．これは繋がれた２つの容器の間を行き来するガスの拡散のモデルである．時間が立つと，ボールは十分に"まざる"のだが，その様子を群論的に調べることができる．この研究では，壺の数を増やし，壺の間に移動の制限を与えるというモデルを環積のゲルファントペアを用いて構成し，拡散の詳細な様子を明らかにした．