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Extending the Steinberg representation. (English) Zbl 0794.20022
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.
Reviewer: Li Fuan (Beijing)

MSC:
20C33 Representations of finite groups of Lie type
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20E42 Groups with a \(BN\)-pair; buildings
20G05 Representation theory for linear algebraic groups
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References:
[1] Bredon, G., Introduction to compact transformation groups, () · Zbl 0246.57017
[2] Curtis, C.W.; Reiner, I., ()
[3] Schmid, P., Rational matrix groups of a special type, Linear algebra appl., 71, 289-293, (1985) · Zbl 0575.20005
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