Development of the algebraic study of graphs

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K05156
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: Tanaka Hajime 東北大学, 情報科学研究科, 准教授 (50466546)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: アソシエーションスキーム / 距離正則グラフ / Terwilliger 代数 / グラフのスペクトル

Outline of Final Research Achievements

I explored applications of the representation theory of non-commutative semisimple matrix algebras, such as the Terwilliger algebra which is defined for each vertex of a graph, and obtained results, for example, about the structure and non-existence of so-called relative designs. In the process of studying applications to quantum probability theory and so on, I found new families of univariate hypergeometric Laurent orthogonal polynomials and bivariate hypergeometric orthogonal polynomials, and described their fundamental properties, such as recurrence relations. Besides, I proved a certain conjecture related to quantum information theory, with the help of a technique from algebraic combinatorics. During the period of the research plan, I also started several other projects related to quantum probability theory and information theory, etc., and I plan to publicize the outcomes whenever they are ready.

Free Research Field

代数的組合せ論

Academic Significance and Societal Importance of the Research Achievements

代数的グラフ理論或いはグラフのスペクトル理論は、情報理論等の工学的分野とも直接関わって発展してきたが、代数的観点からは隣接代数に基づいた「可換」の理論であった。一方本研究は非可換代数である Terwilliger 代数等に基づき新たな応用を開拓するもので、これらの理論を「非可換化」或いは「量子化」する試みであると言える。今回特に重点的に行った量子確率論との連携の研究は実際2009年頃より構想を進めてきており、解析する例の多変数化等について一定の成果を上げることができたのは大きな進展である。