Space pairs of the Grassmann algebra: Unexpected pairs with polynomial growth of the codimensions

Abstract

Let K be a field of characteristic zero and E the infinite dimensional Grassmann algebra over K. We determine the weak polynomial identities for pairs $(E,S)$, where S is a subspace of E such that $S\cup \left\{1\right\}$ generates E. Depending on the structure of S, we divide this analysis into four distinct cases. When $(E,S)$ belongs to the first two types, we show that the codimension sequence is $c_n(E,S)=1$ for all n. On the other hand, for certain pairs of the third type, we prove that $c_n(E,S)=2^{n-1}$, i.e., in this case, the codimension sequence coincides with the ordinary (associative) case. We further investigate certain classes of associative pairs $(E,S)$, exhibiting polynomial growth, particularly those whose codimension sequences correspond to Young diagrams with unbounded numbers of boxes below the first row. These pairs show that an usual characterization of polynomial growth of the codimensions does not hold in the case of associative-space pairs.