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Products in Lambda-Rings PDF

60 Pages·1977·1.491 MB·English
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INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clanfy markings or notations which may appear on this reproduction. 1 The sign or "target" for pages apparently lackmg from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc , is part of the material being photographed, a definite method of "sectioning" the material has been followed. It is customary to begin filming at the upper left hand corner of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy These pnnts are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print In all cases the best available copy has been filmed. University Microfilms International 300 N ZeebRoad Ann Arbor, Ml 48106 8409753 Brennan, Joseph Patrick PRODUCTS IN LAMBDA-RINGS University of Illinois at Urbana-Champaign PH.D. 1984 University Microfilms International SOON ZeebRoad,AnnArbor,MI48106 PRODUCTS IN A-RINGS BY JOSEPH PATRICK BRENNAN B.A., University of Chicago, 1977 THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illionis at Urbana-Champaign, 1984 Urbana, Illinois UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN THE GRADUATE COLLEGE JANUARY 1984 WE HEREBY RECOMMEND THAT THE THESIS BY JOSEPH PATRICK BRENNAN ENTITLED. PRODUCTS IN A-RINGS BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE mr UOCTOR OF PHILOSOPHY ft) trector of Thesis Research W". 2"^gl)^LMZ<3U>u. Head of Department Committee on Final Examination! Chairman 2?i ~ hi - (U<Lv£^ Lue&etrfije. L. /rv,R OiVKnr Ijj^frC- 3Xk> J!lM^ ft* sQsidsbjrti t Required for doctor's degree but not for master's iii PRODUCTS IN X-RINGS Joseph Patrick Brennan Department of Mathematics University of Illinois at Urbana-Champaign, 1984 A method is provided for the practical computation of the coef ficients for a number of structural polynomials of the universal A-ring in one indeterminant. In particular, we derive a method for the computa tion of the inner product extending that of Lascoux. We also introduce a method that reduces the analysis of the pie thy sin, at least for special cases, to solving a system of linear Diophantine equations in a distin- quished region of space. iv Dedication To my parents: Francis Patrick Brennan and Elizabeth Reglna Brennan r- -i* »-t V Acknowledgement . 1 jould like to thank Robert M. Fossum for his encouragement, criticism, guidance, and friendship which made this thesis possible. I would like also to thank Mary P. Freier for her moral support during the researching and writing of this work and 0. Melody Armstrong for her excellent job of typing from my holograph. vi TABLE OF CONTENTS Introduction 1 Chapter 0 3 § 1 Partitions 3 §2 Intervals Diagrams and Tableau 4 §3 Words and the Plactlc monoid 5 §4 Jeu de Taquin 8 §5 The Fundamental Theorem 10 §6 Induction and Restriction 11 §7 The Hook Formula 12 §8 The Wreath Product 13 §9 The Outer Product 14 §10 The Inner Product 15 §11 The Outer Plethysm 16 §12 The Relation to the Representation Theory of GL(n) 19 Chapter 1 21 Chapter 2 26 Chapter 3 37 Bibliography 46 Vita 50

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