Littlewood-Richardson rules for ordinary and projective representations of symmetric groups

196 Pages·1991·6.627 MB·English

by  Mark Masami Shimozono

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Littlewood-Richardson rules for ordinary and projective
representations of symmetric groups
Shimozono, Mark Masami, Ph.D.
University of California, San Diego, 1991
U M I
300 N. Zeeb Rd.
Ann Arbor, MI 48106

NIVERS TY OF CAL FORNIA, SAN DIEGO.
A31822006437529B
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UNIVERSITY OF CALIFORNIA, SAN DIEGO
Littlewood-Richardson Rules
for Ordinary and Projective Representations
of Symmetric Groups
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Mathematics
by
Mark Shimozono
Committee in charge:
Professor Jeffrey B. Remmel, Chair
Professor Edward A. Bender
Professor Adriano M. Garsia
Professor Michael Saks
Professor S. Gill Williamson
1991
11
The dissertation of Mark Shimozono is approved, and it
is acceptable in quality and form for publication on
microfilm:
Chair
University of California, San Diego
1991
iii
TABLE OF CONTENTS
Signature Page.................................................................................................iii
Table of Contents............................................................................................iv
Acknowledgements..........................................................................................vi
Vita and Publications....................................................................................vii
Abstract..........................................................................................................viii
Introduction.......................................................................................................1
I Chapter 1. Representations and the LR rule .............................................4
1.1 Representation theory of finite groups...........................................4
1.2 Representations of symmetric groups ...........................................9
1.3 Projective representations...............................................................18
1.4 Projective representations of symmetric groups.........................24
II Chapter 2. Compatibility..............................................................................38
2.1 Definitions..........................................................................................38
2.2 Schensted insertion..........................................................................44
2.3 Standard Labelling.........................................................................55
2.4 The Slide Lemma and jeu de taquin.............................................63
2.5 Shuffles................................................................................................77
2.6 Insertion and jeu de taquin............................................................80
2.7 Insertion symmetries........................................................................84
2.8 Reading tableaux..............................................................................90
2.9 Compatibility....................................................................................93
2.10 Various compatibilities.................................................................99
III Chapter 3. Shift compatibility...................................................................102
3.1 Shifted Tableaux............................................................................102
3.2 Shifted insertion ............................................................................105
3.3 Labelling and shifted insertion ...................................................117
3.4 Shift compatibility........................................................................122
3.5 Row words and shuffles.................................................................126
3.6 Nonskew shifted shapes and reading tableaux ........................133
3.7 Two row rectangles........................................................................138
3.8 Involutions on generalized shifted tableaux .............................144
IV Chapter 4. Littlewood-Richardson rules and generalizations......148
4.1 Introduction.....................................................................................148
4.2 The Kostka matrix and r-pairing..............................................157
4.3 The inverse Kostka matrix and special rim hook tabloids .. 166
4.4 The classical Littlewood-Richardson rule..................................172
4.5 Bijections.........................................................................................175
Conclusion.....................................................................................................180
Bibliography..................................................................................................181
ACKNOWLEDGEMENTS
I would like to thank my family for their unwavering support in seasons of
frustration and uncertainty as well as in times of victory.
I am indebted to my instructors. Jeff Remmel had a project on hand when
I needed one and was always available to hear both my imprecise babbling and
reasonable ideas.
The combinatorics group at UCSD has been warm, enthusiastic, and sup­
portive. I will miss the fellowship of APM 2250.