Algebraic combinatorics of plane partitions and alternating sign matrices, and related representation theory and mathematical physics

2016 Fiscal Year Final Research Report

• Project/Area Number: 24340003
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Soichi Okada 名古屋大学, 多元数理科学研究科, 教授 (20224016)
• Co-Investigator(Kenkyū-buntansha): 石川 雅雄 琉球大学, 教育学部, 教授 (40243373) ; 松本 詔 鹿児島大学, 理学部, 准教授 (60547553) ; 中西 知樹 名古屋大学, 多元数理科学研究科, 教授 (80227842)
• Project Period (FY): 2012-04-01 – 2017-03-31
• Keywords: 平面分割 / 交代符号行列 / 対称関数 / Pfaffian / KP 階層 / 複素鏡映群

Outline of Final Research Achievements

We study various aspects of algebraic combinatorics related to plane partitions and alternating sign matrices. 1. We establish Pfaffian analogues of the Cauchy-Binet formula and prove several identities among Schur Q-functions and their generalizations. 2. We prove the generating function for Young books can be written as a q-Selberg-type integral. 3. By establishing Pieri rules for classical groups, we give another proof to Burrill's conjecture and find its generalizations. 4. We find Giambelli-type determinant identities for the expansion coefficients of the \tau-function of the KP hierarchy, and prove that these determinant identities characterize solutions of the KP hierarchy.

Free Research Field

組合せ論，表現論