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ON TRANSLATION FUNCTORS FOR GENERAL LINEAR AND SYMMETRIC GROUPS

Published online by Cambridge University Press:  01 January 2000

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Abstract

In this paper we study the rational representation theory of the general linear group $G = \mbox{GL}_n(F)$ over an algebraically closed field $F$ of characteristic $p$. Given $\alpha \in {\Bbb Z} / p{\Bbb Z}$, we define functors $\mbox{Tr}^\alpha$ and $\mbox{Tr}_\alpha$,which, roughly speaking, are given by tensoringwith the natural $G$-module $V$ andits dual $V^*$ respectively, and then projecting ontocertain blocks determined by the residue $\alpha$.In fact, these functors can be viewed as special cases ofJantzen's translation functors.We prove a number of fundamental properties about these functors and also certain closely related functors that arise in the modular representation theory of the symmetric group.

1991 Mathematics Subject Classification: 20G05, 20C05.

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Type
Research Article
Copyright
2000 London Mathematical Society

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