From Grassmann complements to Hodge-duality

Hodge duality is a central concept of 20th century algebraic and analytic geometry and plays a non-negligible role also in recent mathematical physics. At first sight one might expect that its origins lie in the 1930s when its name-giving protagonist, William V.D. Hodge, started his mathematical research. On the other hand, a close link between Hodge's theory and the Maxwell equation has sometimes been claimed not only from a systematic point of view but also historically. In the transformation of classical electromagnetism to the relativistic Maxwell theory the influence of Grassmann's alternating product and an operation he called "complement" played a certain role. In fact, Grassmann's "complements" were a linear algebraic template of the later Hodge star operation. This paper surveys the development from Grassmann's complements and their appearance in relativistic electrodynamics to Hodge's theory of harmonic forms.