Symmetric and exterior powers, linear source modules and representations of Schur superalgebras (Q2766433)

Let \(n\) be a positive integer and let \(k\) be an infinite field. The category of \(k\text{GL}_n(k)\)-modules which are polynomial of degree \(r\) is naturally equivalent to the category of modules for the finite dimensional algebra \(S(n,r)\), known as the Schur algebra and studied extensively by Green. A tensor product of symmetric powers of the natural \(\text{GL}_n(k)\)-module is polynomial of degree \(r\), say, and provided that \(r\leq n\), is injective as an \(S(n,r)\)-module, and all injective indecomposable modules occur as direct summands of such modules. The summands of tensor products of symmetric powers of total degree \(r\) are therefore precisely the injective indecomposable \(S(n,r)\)-modules, and these are parametrized by the partitions of \(r\). Analogously, one may consider modules which are summands of tensor products of exterior powers of the natural module. In this case the indecomposable summands of such modules of total degree \(r\) are exactly the indecomposable (partial) tilting modules and again these are classified by partitions of \(r\), provided \(r\leq n\). The construction and properties of the injective and tilting modules are extremely analogous. Furthermore the two constructions are in duality in the sense that they are interchanged by Ringel duality for the Schur algebra \(S(n,r)\) and the characters of injective and tilting modules are interchanged by the standard involution of symmetric function theory.NEWLINENEWLINENEWLINEIn this paper we consider modules which are summands of a tensor product of symmetric powers and exterior powers of the natural module. We call these `listing modules'. We find that, if \(k\) has characteristic \(p>2\), the summands of such modules, in degree \(m\leq n\), are labelled by pairs \((\lambda,\mu)\), where \(\lambda\) is a partition of \(r\), where \(\mu\) is a partition of \(s\), and \(m=r+ps\). The approach to these summands is via the Schur superalgebras, as described in Muir's thesis, and we count the number of isomorphism types of summands using the version of Green correspondence due to Grabmeier.NEWLINENEWLINENEWLINEIn order to remove the restriction that \(k\) is infinite we shall, where appropriate, use the language of group schemes in dealing with representations of general linear groups.NEWLINENEWLINENEWLINEThe plan of the paper is as follows. In Section 1 we study indecomposable modules for the symmetric group \(\text{Sym}(r)\) of degree \(r\) which occur as summands of modules induced from one dimensional modules for Young subgroups. We call these signed Young modules. We count the number of these using the version of Green correspondence due to J. Grabmeier.NEWLINENEWLINENEWLINEIn Section 2 we study modules over the Schur superalgebra \(S(m|n,r)\). We discuss the Schur functor \(\Phi\colon\text{mod}(S(m|n,r))\to\text{mod}(\text{Sym}(r))\) (generalizing the usual Schur functor), for \(r\leq m,n\). We use this, and the determination of the number of isomorphism classes of signed Young modules of Section 1, to give a natural labelling, by highest weight, of the irreducible modules for the Schur superalgebra. Section 3 is devoted to a study of the listing modules mentioned above, and we give a natural parametrization of these based on the labelling of the irreducible modules for Schur superalgebras in Section 2. In Section 4, we show that the listing modules provide a \(\mathbb{Z}\)-basis of indecomposable functors for the ring of polynomial functors generated by symmetric and exterior powers. We also show that such functors may be interpreted as modules over a suitable associative algebra (the infinite dimensional Schur algebra). We show further that the ring of polynomial functors mentioned above is a free polynomial ring and give free generators. In the final Section we describe the Green correspondents of the signed Young modules, generalizing the analogous result for Young modules due to J. Grabmeier.