Splitting fields and twisted group rings for the finite general linear groups (Q2776265)

Let \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]_\phi\) denote the fixed point algebra under the diagonal action of the Galois group \(\text{Gal}(\mathbb{F}_q:\mathbb{F}_p)\) on the group ring \(\mathbb{F}_q[\text{GL}_n(\mathbb{F}_q)]\). It is a twisted form of the group ring \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]\). The author shows that one could argue it is actually the other way around: The group ring \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]\) is twisted, and \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]_\phi\) is in fact split, i.e. the endomorphism algebra of each simple \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]_\phi\)-module is one dimensional. In contrast, the group ring \(\mathbb{F}_p[\text{GL}_n(\mathbb{F}_q)]\) has \(\mathbb{F}_q\) as minimal splitting field. The proof uses recollement of categories of functors and a kind of Schur-Weyl duality. A similar result is derived for the semigroup ring \(\mathbb{F}_p[\text{M}_n(\mathbb{F}_q)]\), and a connection with Schur algebras is given.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].