Decomposition matrices for exceptional groups at d=4

TL;DR

This paper computes the decomposition matrices for unipotent blocks of certain exceptional groups of Lie type at a specific defect, using advanced cohomological and character-theoretic methods.

Contribution

It provides explicit decomposition matrices for unipotent blocks of exceptional groups at defect , employing new cohomological techniques and properties of Gelfand-Graev characters.

Findings

Decomposition matrices for unipotent -blocks of exceptional groups determined.

New cohomological methods applied successfully to this problem.

Properties of Gelfand-Graev characters confirmed in good characteristic.

Abstract

We determine the decomposition matrices of unipotent $\ell$-blocks of defect $\Phi_4^2$ for exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first author, together with properties of generalized Gelfand-Graev characters which were recently shown to hold whenever the underlying characteristic is good.