Letter-Place Methods and Homotopy

Abstract

In this paper I want to illustrate the use of letter-place methods in the construction of homotopies for resolutions of certain Weyl modules. I’ll look at two cases: the resolutions of two-rowed skew-shapes (already in the literature [B-R 1]), and the resolutions of hooks (work in progress; only the first three stages of the homotopy appear here). The desire to construct such homotopies arose from the attempt to answer Rota’s question: What do the syzygies of these modules look like? Since, in representation theory and combinatorics, the ‘correct’ form of an answer to such a question is in terms of a basis consisting of easily described tableaux, the double standard tableaux that appear in letter-place algebras lend themselves admirably to this topic.