Study on random partitions and random matrices based on combinatorics and representation theory

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05281
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kagoshima University
• Principal Investigator: Matsumoto Sho 鹿児島大学, 理工学域理学系, 准教授 (60547553)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 対称群 / ヤング図形 / 既約指標 / ランダム行列 / Weingarten calculus

Outline of Final Research Achievements

As a study on random matrices, especially on the Weingarten calculus, we have investigated the distribution of its partial traces for uniformly distributed random Hermitian matrices on tensor products. For random partitions, we have obtained Kerov polynomial representations and Stanley character formulas for the normalized spin irreducible characters of the symmetric group, and used them to obtain the law of large numbers and the central limit theorem for random shifted Young diagrams.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

ランダム行列論はその名の通りランダムに振る舞う行列の集団を扱う分野で、数理物理学や量子情報理論を始め広い範囲に応用されている。また、その離散への類似物として、整数の分割をランダムに扱う、ランダム分割の研究がある。本研究はこれらについて具体的で視覚的に分かりやすい結果を得た。