2018 Fiscal Year Final Research Report
Plane partitions from the viewpoint of biorthogonal polynomials
| Project/Area Number |
16K05058
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Kyoto University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2016-04-01 – 2019-03-31
|
| Keywords | 平面分割 / 直交多項式 / 数え上げ組合せ論 / 可積分系 |
|---|
| Outline of Final Research Achievements |
A plane partition is a two-dimensional array of non-negative integers which weakly decrease along each row and column. In this study a method is formulated to find nice (product) formulas for (reverse) plane partitions from biorthogonal polynomials and from solutions to the discrete two-dimensional (2D) Toda lattice, that is a dynamical system associated with biorthogonal polynomials. New nice formulas which generalizes some of the existing generating functions for (reverse) plane partitions, such as MacMahon's generating function and the trace generating function, are derived from a specific family of biorthogonal polynomials and from a specific solution to the discrete 2D Toda lattice.
|
| Free Research Field |
数え上げ組合せ論
|
| Academic Significance and Societal Importance of the Research Achievements |
組合せ論的オブジェクトである平面分割は，それ自身に対する研究のみならず，数学の他分野に関連付けられたり物理学などの他領域に応用されたりしている．その背景には積型のよい母関数の数学的な扱いやすさがある．平面分割のよい母関数の存在は応用も含めて重要であるが，よい母関数を系統的に見つけるための処方箋はこれまでなかった．本研究の意義は，双直交多項式などの他分野の道具をうまく用いることで，平面分割のよい母関数を系統的に構成するための手続きを作り出した点にある．本研究で生み出した新しい母関数は，平面分割への理解を深めるだけでなく将来的に他分野への応用にも繋がると期待される．
|
