Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen
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592 Detlef Voigt
Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen
Springer-Verlag Berlin· Heidelberg· New York 1977
Lecture Notes in Mathematics Edited by A Dold and B. Eckmann
592 Detlef Voigt
Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen
Springer-Verlag Berlin· Heidelberg· New York 1977
Author Detlef Voigt Fakultat fur Mathematik Universitat Bielefeld Universitatsetrafre 4800 Bielefeld l/BRD
Library of Congress Cataloging in Publication Data
Voigt, Detlef, 1938Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen. (Lecture notes in mathematics ; 592) Bibliography: p. Includes index. 1. Representa.tions of groups. 2. Finite groups. I. Title. II. Series: Lecture notes in mathematics (Berlin) : 592. QA3.L28 no. 592 [Q,AJ.71) 510' .8s [512' .2) 77-9895
AMS Subject Classifications (1970): 14L05, 14L20, 16A24, 16A64, 16A68,17B50 ISBN 3-540-08251-4 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-08251-4 Springer-Verlag New York' Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Spnnger-Verlag Berlin' Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
v 0 R W0 R T Wir betrachten zwei endliche, algebraische Gruppen G'e G Uber dem algebraisch abgeschlossenen Grundkorper k mit der Charakteristik p>O sowie einen endlich-dimensionalen G'-Modul M. Die den Gruppen zugeordneten Gruppenalgebren seien wie Ublich durch H(G')CH(G) bezeichnet. In der vorliegenden Arbeit sollen einige Fragen untersucht werden, die die Struktur des induzierten G-Moduls H(G) G9 M H(Gt) betreffen. Die beiden wichtigsten Ergebnisse sind die Satze (-l
k[X . .) /J.k[X . . ) 1.J det 1,J det
given by the equation: (j)(f)
=
f(k[X . . ) /J·k[X . . J )(CX:-:)) , 1,J 1.J det 1.J det
denotes again the residue is the inverse of,!,. In this equation 1,J class of X. . in k[X . . ] [J·k[X . .) 1.J 1.J det 1.J det
The observations made above in 0.10. especially yield the important fact. that the function algebra 0(G) of the affine. algebraic group G determines G considered as a set valued functor up to isomorphism. In fact. if we denote for any two k-algebras R, S E Mk the set of k-algebra homomorphisms from R to S by Mk(R,S), then we obtain because of 0.10. the canonical bijective mapping: MkCtJ(G), S)
Mk(k[Xi,J']
det
/J.k[X . . J , S) 1.J det
G(S)
which is functorial in S E Mk. 0.12. Let us now consider the functors of the form Mk(R,?) more exactly. For R E Mk we shall denote the functor Mk(R.?) also by SPk(R) or. if no confusion is possible, by SpeR
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