Geometry of tropical variety and homogeneous spaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610008
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Aoyama Gakuin University
• Principal Investigator: Nishiyama Kyo 青山学院大学, 理工学部, 教授 (70183085)
• Co-Investigator(Kenkyū-buntansha): Iwao Shinsuke 青山学院大学, 理工学部, 助教 (70634989)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 戸田格子 / Lax行列 / totally positive matrix / ヤコビ多様体 / 超離散化 / トロピカル幾何 / 量子コホモロジー / シューベルト多項式

Outline of Final Research Achievements

We studies the methods of (ultra-)discretization of Toda lattice through the Lax formalism. There are so many researches in this field, but our emphasis is on the totally nonnegative matrices and algebro-geometric tools, such as singular (nodal) rational curves and Jacobian varieties. Thus our main result is proving the Lax matrices of Toda lattice which is totally nonnegative give the totally positive part of the Jacobian variety, which is summarized into the paper: Geometric interpretation of the totally nonnegative part of the finite Toda lattice via a singular rational curve. The paper is now being prepared.
We also studied on the quantum cohomology on the flag varieties and Peterson isomorphism. In this direction, we obtained various explicit formulas of Schubert polynomials and/or Grothendieck polynomials, especially their determinantal expressions. However, the results are somewhat rambling and it seems it will take some more time to organize them in a nice way.

Free Research Field

表現論