On Relatively Projective Modules and the Külshammer–Robinson Basis

Abstract

In this paper, we rewrite the relations between the Külshammer–Robinson basis and Navarro basis in the local blocks, under Brauer correspondence. We define N-numbers and KR-matrix, then prove that they can be decided by decomposition numbers and Cartan matrix of local blocks. We give explicit connections between Külshammer–Robinson basis and projective modules. For N-blocks, we give other ways to describe the connections.

Key Words:

• Projective modules
• Characters
• Blocks

Acknowledgments

The work was done when the author was supported by CSC, studying in the School of Mathematics and Statistics, University of Birmingham, under the supervision of Professor G. R. Robinson. I thank Professor G. R. Robinson for all his help. I also thank Dr. C. Eaton for many discussions, and the School of Mathematics and Statistics, University of Birmingham, for offering me a good working environment. The author is also grateful for the referee's comments.

Research partially supported by CNSF and CSC.