Hall polynomials for classical groups

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