Virtual Workshop on Recent Developments in Geometric Representation Theory

For a finite group $G$ one has a process of modular reduction which takes a $KG$-module, over a field $K$ of characteristic zero, and produces a $kG$-module, over a field $k$ of positive characteristic. Starting with a simple $KG$-module its modular reduction may fail to be simple. The rows and columns of the decomposition matrix are labelled by the simple $KG$-modules and simple $kG$-modules respectively. The rows of the matrix encode the multiplicities of the simple $kG$-modules in a composition series of the corresponding reduced $KG$-module.

The decomposition matrix can be partitioned with respect to the blocks of $kG$. When $G = G(\mathbb{F}_q)$ is a finite reductive group, such as the finite general linear group, the unipotent blocks form a prototype for all blocks. In this talk we will present joint work with O. Brunat and O. Dudas showing that for some ordering of the rows and columns the decomposition matrix of the unipotent blocks has a lower unitriangular shape (under some mild conditions on $q$ and the characteristic of $k$).