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.
Plamen Koshlukov, Department of Mathematics, IMECC, UNICAMP, P.O. Box 6065, 13083-970 Campinas, SP, Brazilplamen@ime.unicamp.br
Louwsma, Joel; Presoto, Adilson Eduardo; and Tarr, Alan
"A New Computation of the Codimension Sequence of the Grassmann Algebra,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 6
, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol6/iss2/7