The affine Coxeter transformation
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R. Stekolshchik
Notes on Coxeter Transformations and the McKay Correspondence
Rafael Stekolshchik Str. Kehilat Klivlend 7 Tel-Aviv Israel [email protected]
ISBN 978-3-540-77398-6
e-ISBN 978-3-540-77399-3
DOI 10.1007/978-3-540-77399-3 Springer Monographs in Mathematics ISSN 1439-7382 Library of Congress Control Number: 2007941499 Mathematics Subject Classification (2000): 20F55, 15A18, 17B20, 16G20 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Summary
We consider the Coxeter transformation in the context of the McKay correspondence, representations of quivers, and Poincar´e series. We study in detail the Jordan forms of the Coxeter transformations and prove splitting formulas due to Subbotin and Sumin for the characteristic polynomials of the Coxeter transformations. Using splitting formulas we calculate characteristic polynomials of the Coxeter transformation for the diagrams T2,3,r , T3,3,r , T2,4,r , prove J. S. Frame’s formulas, and generalize R. Steinberg’s theorem on the spectrum of the affine Coxeter transformation for the multiplylaced diagrams. This theorem is the key statement in R. Steinberg’s proof of the McKay correspondence. For every extended Dynkin diagram, the spectrum of the Coxeter transformation is easily obtained from R. Steinberg’s theorem. In the study of representations πn of SU (2), we extend B. Kostant’s construction of a vector-valued generating function PG (t). B. Kostant’s construction appears in the context of the McKay correspondence and gives a way to obtain multiplicities of irreducible representations ρi of the binary polyhedral group G in the decomposition of πn |G. In the case of multiply-laced graphs, instead of irreducible representations ρi we use restricted representations and induced representations of G introduced by P. Slodowy. Using B. Kostant’s construction we generalize to the case of multiply-laced graphs W. Ebeling’s theorem which connects the Poincar´e series [PG (t)]0 and the Coxeter transformations. According to W. Ebeling’s theorem [PG (t)]0 =
X (t2 ) , X˜ (t2 )
where X is the characteristic polynomial of the Coxeter transfo
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