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University
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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
