Extending the Steinberg representation
Let \(G\) be a finite group with a split \(BN\)-pair of rank \(n\geq 1\) and characteristic \(p\), and let \(St_ G\) be its Steinberg character. The author [Linear Algebra Appl. 71, 289-293 (1985; Zbl 0575.20005)] has shown that the Steinberg representation can be extended to any group \(E\) containing \(G\) as a normal subgroup. In the present paper the author gives a more natural approach which is based on a nice description of the Steinberg module. The key proposition provides a \(\mathbb{Q} G\)-homomorphism whose kernel affords the Steinberg character.
scientific article; zbMATH DE number 63905