Computation of boundary components of fundamental domains of symmetric cones and its application

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400004
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yamagata University
• Principal Investigator: Hayata Takahiro 山形大学, 大学院理工学研究科, 准教授 (50312757)
• Research Collaborator: ODA Takayuki 沖縄科学技術大学院大学, 教授 (10109415) ; WATANABE Takao 大阪大学, 理学研究科, 教授 (30201198)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: エルミート定数 / 基本領域 / ジーゲル上半空間 / 最小ベクトル / キス数

Outline of Final Research Achievements

In this research, we propose an algorithm computing the kissing number of an element of classical groups and apply them to concrete cases. This algorithm uses the evaluation of determinants of the positive definite symmetric matrices in the Minkowski domain and so-called the short vector algorithm. The height of the linear transformation is by definition the modified ratio of the covolume of the transformed sublattice in the isotropic space and the covolume of the whole fixed lattice. The density is the minimum value of the height among modular transformation of the lattice and the kissing number is its cardinality modulo certain modular transformations. These are a generalization of the classical sphere packing problem. An application is when the symplectic group of degree 2, matrix size 4 and when the totally isotropic space is chosen. In this case, there
are three kinds of symplectic lattices who have locally maximal kissing numbers.

Free Research Field

数論