P-decomposition matrix of derived Chevalley group

Abstract

Character theory has diverse applications in the applied sciences. In particular, it plays a crucial role in understanding symmetries and features of quantum systems in quantum mechanics, constructing error-correcting codes in coding theory, and analyzing and predicting molecular structures in computational chemistry. In addition, the characters of finite groups are utilized in algebraic geometry, graph theory, cryptography, number theory, network analysis, and cohomology theory. This article studies the derived Chevalley group $G_2'(2)$, which is isomorphic to the group of all $3 \times 3$ invertible matrices preserving a non-singular Hermitian form over the Galois field $\mathbb{F}_9$ of order 9. The computation of p-decomposition matrices relative to all prime divisors of the group order enables the investigation of the decomposition matrices of its subgroups and supergroups. Absolutely irreducible p-characters provide valuable insights into the structure of the group.