Introduction .................................................. vii
Chapter 1. Schubert Bases of Gr and Symmetric Functions ........ 1
1.1 Symmetric functions ........................................ 1
1.2 Schubert bases of Gr ....................................... 1
1.3 Schubert basis of the affine flag variety .................. 3
Chapter 2. Strong Tableaux ..................................... 5
2.1  n as a Coxeter group ...................................... 5
2.2 Fixing a maximal parabolic subgroup ........................ 6
2.3 Strong order and strong tableaux ........................... 6
2.4 Strong Schur functions ..................................... 9
Chapter 3. Weak Tableaux ...................................... 11
3.1 Cyclically decreasing permutations and weak tableaux ...... 11
3.2 Weak Schur functions ...................................... 12
3.3 Properties of weak strips ................................. 13
3.4 Commutation of weak strips and strong covers .............. 14
Chapter 4. Affine Insertion and Affine Pieri .................. 19
4.1 The local rule Øu,υ ........................................ 19
4.2 The affine insertion bijection Φu,υ ........................ 19
4.3 Pieri rules for the affine Grassmannian ................... 24
4.4 Conjectured Pieri rule for the affine flag variety ........ 25
4.5 Geometric interpretation of strong Schur functions ........ 26
Chapter 5. The Local Rule Øu,υ ................................. 27
5.1 Internal insertion at a marked strong cover ............... 27
5.2 Definition of Øu,υ ......................................... 29
5.3 Proofs for the local rule ................................. 29
Chapter 6. Reverse Local Rule ................................. 39
6.1 Reverse insertion at a cover .............................. 39
6.2 The reverse local rule .................................... 41
6.3 Proofs for the reverse insertion .......................... 41
Chapter 7. Bijectivity ........................................ 49
7.1 External insertion ........................................ 50
7.2 Case A (commuting case) ................................... 51
7.3 Case В (bumping case) ..................................... 51
7.4 Case С (replacement bump) ................................. 52
7.1 iv CONTENTS
Chapter 8. Grassmannian Elements, Cores, and Bounded
Partitions ......................................... 55
8.1 Translation elements ...................................... 55
8.2 The action of n on partitions ............................ 58
8.3 Cores and the coroot lattice .............................. 58
8.4 Grassmannian elements and the coroot lattice .............. 60
8.5 Bijection from cores to bounded partitions ................ 60
8.6 fc-conjugate .............................................. 61
8.7 From Grassmannian elements to bounded partitions .......... 61
Chapter 9. Strong and Weak Tableaux Using Cores ............... 63
9.1 Weak tableaux on cores are k-tableaux ..................... 63
9.2 Strong tableaux on cores .................................. 64
9.3 Monomial expansion of t-dependent k-Schur functions ....... 66
9.4 Enumeration of standard strong and weak tableaux .......... 68
Chapter 10. Affine Insertion in Terms of Cores ................ 73
10.1 Internal insertion for cores .............................. 73
10.2 External insertion for cores (Case X) ..................... 74
10.3 An example ................................................ 74
10.4 Standard case ............................................. 75
10.5 Coincidence with RSK as n → ∞ ............................. 76
10.6 The bijection for n = 3 and m = 4 ......................... 77
Bibliography ................................................... 81
|
