Promotion Operators in Representation Theory and Algebraic Combinatorics

Abstract
iv
Acknowledgments
v
Introduction
1
Chapter 1.
Basic Tableaux Combinatorics
4
1.1.
Young tableaux
4
1.2.
The plactic monoid
8
1.3.
Standardization
15
1.4.
Promotion
17
Chapter 2.
A promotion operator on rigged configurations of type A
21
2.1.
Introduction
21
2.2.
Preliminaries and the main result
22
2.3.
Outline of the proof of the main result
43
2.4.
Proof of Proposition 2.3.3
51
2.5.
Proof of Proposition 2.3.5
53
2.6.
Proof of Proposition 2.3.7
55
2.7.
Proof of Proposition 2.3.10
59
2.8.
Proof of Proposition 2.3.14
61
2.9.
Proof of Proposition 2.3.15
64
Appendix 2.A.
Proof of Proposition 2.8.5
69
Appendix 2.B.
Several useful facts
71
Chapter 3.
Promotion and evacuation on rectangular and staircase tableaux
74
3.1.
Introduction
74
3.2.
Definitions and Preliminaries
76
3.3.
The embedding of SY T(sck) into SY T(k(k+1))
79
3.4.
Descent vectors
81
3.5.
Some comments and questions
86
Appendix 3.A.
Proof of Lemma 3.4.10
89
Chapter 4.
The commutativity between the R-matrix and the promotion operator – a combinatorial
proof
91
4.1.
A combinatorial algorithm for R
91
4.2.
Interaction between ρ and bumping
97
4.3.
The proof of the commutativity
100
Appendix 4.A.
Proof of Lemma 4.3.7
106
Bibliography
109