Modular Representations and the Decomposition of Induced Characters: A Geometric Approach

This paper explores the intricate relationship between modular representations of finite groups and the decomposition of induced characters, employing a geometric perspective. We investigate how the geometric structure of algebraic varieties associated with group actions influences the decomposition matrices arising in modular representation theory. Specifically, we analyze the decomposition of induced characters from subgroups to the full group in the context of a finite group's representations over a field of characteristic p. We introduce geometric tools such as the Frobenius morphism and étale covers to study the congruences between ordinary characters and modular characters. The objective is to provide a framework that uses geometric insights to enhance our understanding of the representation theory of finite groups in positive characteristic, shedding light on open problems related to the block structure and the computation of decomposition numbers. We present original results concerning the behavior of decomposition matrices under certain geometric conditions, offering a novel approach to the study of modular representations.