Hall polynomials for classical groups

✍ Scribed by Franklin Miller Maley

• Year: 1996
• Tongue: English
• Leaves: 97
• Series: PhD thesis at Princeton University
• Category: Library

✦ Table of Contents

Abstract .................................................................................................... iii
1: Ordinary Hall Polynomials ............................................................. 1
1.1: Background ................................................................................. 1
1.2: The category of p a irs.................................................................4
1.3: Reduction to F-polynomials ..................................................... 8
1.4: A new formula for F-polynomials ........................................... 10
2: Symplectic and Orthogonal Groups ............................................... 16
2.1: Definition of Hall functions ................................................... 16
2.2: Unipotents versus nilpotents ................................................... 21
2.3: Nilpotent conjugacy classes ................................................... 23
2.4: The Springer-Steinberg basis ................................................... 26
3: Computing General Hall Polynomials ........................................... 30
3.1: Overview of the calculation ................................................... 30
3.2: Action of the centralizer ........................................................... 32
3.3: Details of the calculation ...................................................... 39
3.4: Details of the recursion .......................................................... 43
4: Main Results ................................................................................... 46
4.1: Dependence on q mod 4 .......................................................... 46
4.2: Integrality and Fourier transforms ....................................... 52
4.3: The one-block c a s e ................................................................... 59
4.4: Directions for further s tu d y ................................................... 63
5: Unitary Hall Polynom ials ............................................................... 65
5.1: Classes and o r b its ................................................................... 66
5.2: Computations involving unitary g ro u p s .............................. 68
5.3: The one-block c a s e .................................................................. 69
References ............................................................................................... 71
A ppendix ............................................................................................... 73