Krakowski and Regev found a basis of polynomial identities satisfied by the Grassmann algebra over a field of characteristic $0$ and described the exact structure of these relations in terms of the symmetric group. Using this, they found an upper bound for the the codimension sequence of the $T$-ideal of polynomial identities of the Grassmann algebra. Working with certain matrices, they found the same lower bound, thus determining the codimension sequence exactly. In this paper, we compute the codimension sequence of the Grassmann algebra directly from these matrices, thus obtaining a proof of the codimension result of Krakowski and Regev using only combinatorics and linear algebra. We also obtain a corollary from our proof.

Author Bio

Joel Louwsma is an undergraduate mathematics major at the University of Michigan. In July 2004 he participated in an REU at UNICAMP, Brasil, where the research described in this paper was conducted. He plans to graduate in May 2005 and begin graduate school in mathematics the following fall.