关于有限群的p-块的数值不变量的若干研究

In modular representation theory of finite groups, one of the most important tasks is to determine numerical invariants of p-blocks , including the number of irreducible (ordinary /Brauer) characters and the Cartan invariants of p-blocks. In the present project, we will launch our research from the following two aspects: first, with the help of integral quadratic forms, we will estimate the number of irreducible ordinary characters in a p-block B in terms of Cartan invariants of B-subsections, and verify Brauer’s k(B)-Conjecture and Olsson’s Conjecture on this basis; second, we will investigate the properties of lower defect group multiplicities and subpair multiplicities associated to a p-block B, and use these tools to determine k(B), l(B) and the elementary divisors of the Cartan matrix of B.

在有限群的模表示论中，最重要的研究课题之一是计算p-块的各种数值不变量，其中包括p-块中不可约(常/Brauer)特征标的个数以及p-块的Cartan不变量。本项目拟从以下两个方面展开研究：其一，以整系数二次型为工具，通过分析有限群的p-块的子部的Cartan矩阵，估计p-块中不可约常特征标的个数，并在此基础上验证Brauer的k(B)-猜想和Olsson猜想；其二，探讨下亏群重数和子对重数的性质，并以它们为工具计算p-块中不可约 (常/Brauer)特征标的个数以及Cartan矩阵的初等因子。