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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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