Characteristic-free representation theory of the general linear group (Q1085277)

In an earlier paper [Adv. Math. 44, 207-278 (1982; Zbl 0497.15020)] the authors and \textit{J. Weyman} developed a theory of Schur functions and complexes over a commutative ring R. These ideas are used in this paper to study the polynomial representations of GL(F), where F is a finitely generated free R-module. If \(\phi\) : \(G\to F\) is a map of finitely generated free R-modules, a Schur complex \(L_{\alpha}(\phi)\) is defined for any shape \(\alpha\). (Here a shape is a matrix with zeros and ones as entries, and generalizes a skew partition.) In particular, maps of the form \(0\to F\) give rise to Schur functors \(L_{\alpha}(F)\), which are GL(F)-modules. These concepts are reviewed in {\S} 2. In {\S} 3, the Pieri formulas for \(L_{\lambda}F\otimes S_ pF\) and \(L_{\lambda}F\otimes \Lambda^ pF\) (where \(\lambda\) is a partition, and \(S_ p\), \(\Lambda^ p\) denote symmetric and exterior powers) are derived over R. In {\S} 4, for a skew partition \(\alpha =\lambda /\mu\) where \(\lambda\), \(\mu\) have two parts, a resolution \({\mathbb{C}}(\alpha)\) for \(L_{\alpha}F\) over R by direct sums of tensor products of exterior powers of F is obtained. As an application \(\Lambda^ 2(\Lambda^ 2F)\) is calculated over a field of characteristic p. In {\S} 5 the authors construct an exact sequence \[ 0\to L_{\lambda (m)}F\to...\to L_{\lambda (1)}F\to L_{\lambda}F\to 0 \] whose terms \(L_{\lambda (i)}F\) are Schur functors which can be explicitly described, for a partition \(\lambda\). An analogous construction is given for a skew partition in {\S} 6. CoSchur functors are considered in {\S} 7 and finally some forthcoming results are described in {\S} 8.