Aspects of higher degree forms with symmetries

• Main Author: Omar, Mohammed Rafiq
• Other Authors: Hughes, Kenneth R
• Format: Doctoral Thesis
• Language: English
• Published: University of Cape Town 2016
• Subjects: Mathematics and Applied Mathematics
• Online Access: http://hdl.handle.net/11427/16109

Bibliography: pages 113-119. === In Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant.