Classification of Exterior Forms

Karve, Vaibhav (2015) Classification of Exterior Forms. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Vaibhav Karve (10MS19)) Vaibhav_Karve_10MS19_MSThesis.pdf - Submitted Version Restricted to Repository staff only Download (523kB)

Abstract

The space of exterior forms (also called alternating multilinear forms, k-forms, or k-blades) has a very rich and interesting algebraic structure. Chief among its properties to be studied historically has been the classification problem of exterior forms upto pullback. This classification has been carried out on real, complex and finite fields to great degrees of success and has been the prime motivator for defining and studying important notions such as divisibility, rank and stability. While the classification result itself is easy to understand, its proof that can be found in existing mathematical literature is not. The proof employs difficult arguments in Galois cohomology and is therefore inaccessible to those who are not experts in algebra. Moreover, the existing solution is quite old and uses notation and terminology that is inconsistent with modern trends. This thesis presents an alternate proof of the classification upto pullback of three-forms on a six dimensional real vector space, which I claim is many times simpler than the existing solution. My proof depends heavily on the division algorithm, which provides a way of extracting a \quotient" and \remainder" on factoring out a one-form from a k-form. The division algorithm also reveals an inherent structure in the orbits thus obtained that was overlooked by the old proof. Moreover, my proof shows reasonable promise of being generalizable to seven and eight dimensions.

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