Whitney algebras and Grassmann's regressive products

Geometric products on tensor powers $\Lambda(V)^{\otimes m}$ of an exterior algebra and on Whitney algebras \cite{crasch} provide a rigorous version of Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge $\ast-$operators and the {\it join} and {\it meet} products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings of tensor powers $\Lambda(V)^{\otimes m}$ and of Whitney algebras $W^m(M)$ in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley-Grassmann algebras.