Representing Finite Groups: A Semisimple Introduction

Representing Finite Groups Ambar N. Sengupta Representing Finite Groups A Semisimple Introduction 123 Ambar N. Sengupta Department of Mathematics Louisiana State University Baton Rouge Louisiana USA [email protected] ISBN 978-1-4614-1230-4 e-ISBN 978-1-4614-1231-1 DOI 10.1007/978-1-4614-1231-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941437 Mathematics Subject Classification (2010): 20C05, 20C30, 20C35, 16D60, 51F25 c Springer Science+Business Media, LLC 2012  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my mother Preface Geometry is nothing but an expression of a symmetry group. Fortunately, geometry escaped this stifling straitjacket description, an urban legend formulation of Felix Klein’s Erlangen program. Nonetheless, there is a valuable ge(r)m of truth in this vision of geometry. Arithmetic and geometry have been intertwined since Euclid’s development of arithmetic from geometric constructions. A group, in the abstract, is a set of elements, devoid of concrete form, with just one operation satisfying a minimalist set of axioms. Representation theory is the study of how such an abstract group appears in different avatars as symmetries of geometries over number fields or more general fields of scalars. This book is an initiating journey into this subject. A large part of the route we take passes through the representation theory of semisimple algebras. We will also make a day-tour from the realm of finite groups to look at the representation theory of unitary groups. These are infinite, continuous groups, but their representation theory is intricately interlinked with the representation theory of permutation groups, and hence this detour from the main rout