Operators on k-tableaux and the k-Littlewood-Richardson rule for a special case

Contents ii
List of Figures iv
List of Tables v
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Where k-Schur functions came from . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Strategies used to prove the k-Littlewood–Richardson rule . . . . . . . . . . 3
2 Background 5
2.1 Partitions, tableaux, k + 1-cores . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Symmetric functions and Schur functions . . . . . . . . . . . . . . . . . . . . 9
2.3 k-tableaux and k-Schur functions . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Properties of entries in k-tableaux . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Some operators on k-tableaux 19
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Properties of the operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 The operator s (k)
i
e (k)
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 The k-lattice property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 A proof of the k-Littlewood–Richardson rule in a restricted case 30
4.1 The proof of the k-Littlewood–Richardson rule . . . . . . . . . . . . . . . . . 30
4.2 An example of the bijection used for the proof . . . . . . . . . . . . . . . . . 33
4.3 An explanation of the requirements on µ and problems with generalizing the
proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 An alternate proof of the k-Littlewood–Richardson rule for µ = (a,b) 37
5.1 The alternate proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Problems with generalizing the classical case and an example . . . . . . . . . 41
6 A strategy for computing k-Littlewood–Richardson coefficients 44
7 The scalar product of a dual k-Schur function and a Schur function 47
Bibliography 50