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Hall-Littlewood Vertex Operators and the Kostka-Foulkes Polynomials

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Author(s): Timothee William Bryan
Series: PhD thesis at North Carolina State University
Year: 2016

Language: English

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Partitions and Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2 Symmetric Functions and the Hall-Littlewood Inner Product . . . . 14
2.1 Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Symmetric Function Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Hall-Littlewood Functions and Kostka-Foulkes Coefficients . . . . . . . . . . . . . 17
2.4 An Explicit Algebraic Formula for the Kostka-Foulkes Polynomials . . . . . . . . 20
Chapter 3 Littlewood-Richardson Tableaux . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Littlewood-Richardson Cores and Tableaux . . . . . . . . . . . . . . . . . . . . . 29
3.2 Littlewood-Richardson Tableaux Lattices . . . . . . . . . . . . . . . . . . . . . . 34
3.3 T-polynomials and t-analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Shifted Littlewood-Richardson Tableaux and Lattices . . . . . . . . . . . . . . . . 51
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62