Some Remarks on Filtrations and Plethysms

Plethysm, introduced by Littlewood, is an operation on symmetric functions that corresponds to composition of Schur and or Weyl functors. Let R be a commutative ring with identity and F a free R-module of finite rank. We denote by 4F, SF, and DF the exterior, symmetric, and divided power algebra of F, respectively. Over the last decade or so, a number of authors have studied plethysms of type Ak(B2 F ), where A, B # [4, S, D], mostly in connection with invariant theory, resolutions of determinantal ideals, and the characteristic-free representation theory of the general linear group (see [A, ADF, B, K1, K2, RS1, RS2]). In a recent article [B], Boffi shows that of the nine plethysms A(B2F ) with A, B # [4, S, D], only four have universal filtrations and he gives explicit descriptions of these. (A ``universal'' filtration for us is one defined over any ring whose composition factors are all Schur or all Weyl modules.) In this note we are interested in filtrations associated to the plethysms A2(BkF ), A, B # [4, S, D]. If R is a field of characteristic zero, the irreducible decomposition of each A2(BF) is known [L, Part II]. However, from [AB1] and [B] it follows that no A2(BF), with A, B # [4, S, D], has a universal filtration. All the countrexamples contained in [AB1] and [B] involve rings where 2 is not invertible. Hence we assume from now on that 2 is invertible in R. In this case, the second divided power functor is naturally isomorphic to the second symmetric power, and therefore there are only six distinct plethysms of type A2(BF), A, B # [4, S, D]. The purpose of this article is to show (by providing explicit filtrations) that each of