On projective and injective polynomial modules

Abstract

Let k denote a field. For each n ≥ 0 we have a category Pol(n) of finite dimensional polynomial modules and a category Rat(n) of rational modules over k. For r ≥ 0, we write Pol(n, r) for the full subcategory of Pol(n) whose objects are the polynomial modules of degree r. Each V ∈ Pol(n) has a unique decomposition V = ⊕∞ r=0 V (r), where V (r) is polynomial of degree r. Furthermore, the category Pol(n, r) is naturally equivalent to the category of modules for the (finite dimensional, associative) Schur algebra S(n, r). It follows that every polynomial module has a projective cover and an injective hull. We here attempt to describe those polynomial modules which are both projective and injective. We give a complete description in the cases n = 2, 3 and make a conjecture which, if true, gives a complete description in the general case. A finite dimensional polynomial module which is injective and projective must be a tilting module and, thanks to a natural duality on the category of polynomial modules, our problem is equivalent to that of determining which