Computations in generic representation theory: maps from symmetric powers to composite functors

Abstract

With this article, the author continues his investigation of``generic'' representation theory, i.e., of the category\scr F(q), with objects the functors from the category offinite dimensional \bold F\sb q vector spaces to the categoryof all \bold F\sb q vector spaces and with morphisms thenatural transformations, where \bold F\sb q is the finite fieldof order q and characteristic p [Amer. J. Math. 116 (1994),no. 2, 327 360; MR 95c:55022; K Theory 8 (1994), no. 4, 395428; MR \Cite{Kuhn94:Generic:395--428}[95k:55038]; K Theory 9(1995), no. 3, 273 303; MR 97c:55026].\par Let S\sb n andS\sp n denote the objects of \scr F(q) defined byS\sb n(V)=(V\sp {\otimes n})\sp {Σ\sb n} andS\sp n(V)=(V\sp {\otimes n})/ {Σ\sb n}, whereΣ\sb n is the symmetric group on n letters. For anyobject F of \scr F(q), let M\sb F denote the graded vectorspace \bigoplus \sb {n=0}\sp {\infty}\roman {Hom}\sb {\scr F(q)}(S\sb n, F). In fact, as the author explains, M\sb F is evenan unstable \scr A(q) module, where \scr A(q) is the Hopfalgebra of Steenrod reduced qth powers. To see this, observethat the category \scr U(q) of unstable \scr A(q) modules isequivalent to the category of additive functors from the fullsubcategory of \scr F(q) with objects the S\sp n's to thecategory of \bold F\sb q vector spaces.\par The author's goalin this paper is to explain how to compute M\sb {F\circ G} interms of M\sb G for various functors F. First, for any objectF in \scr F(q), he constructs a functorU\sb F\colon \scr U(q)@>>>\scr U(q) such that U\sb F(M\sb G)is naturally isomorphic to M\sb {F\circ G}, for all functorsG that are locally finite. A functor is locally finite if it isequal to the union of all of its subfunctors that have finitecomposition series such that the subquotients have no nontrivialsubobjects.\par The author then studies objects F of\scr F(q) that extend naturally to endofunctors of \scr U(q):S\sb n, multiples of the nth tensor power functor, T\sp n,by any element of the group ring \bold F\sb q(Σ\sb n),\S\sp n modulo pth powers, denoted \overline S{}\sp n, andthe exterior power functor Λ\sp n. He proves that forthese objects, F(M\sb G) is naturally isomorphic toM\sb {F\circ G}. In particular, M\sb {F\circ G} depends onlyon the vector space structure of M\sb G.\par The authorestablishes interesting results for two other important types ofobjects in \scr F(q) as well. For any \bold F\sb q vectorspace W, let J\sb W denote the injective object of\scr F(q) defined by J\sb W(V)=\bold F\sb q\sp {\roman {Hom}(V,W)}. In the case when q is a prime, theauthor provides an explicit description of M\sb {J\sb W\circ G}and M\sb {S\sp n\circ G} in terms of M\sb G, obtaining as aconsequence that their Poincare series depend only on thePoincare series of M\sb G.\par In proving the above mentionedresults the author relies on his previous study of \scr F(q)and \scr U(q) in the three papers cited above. In particular,his embedding and vanishing theorems are key elements of theproofs in this article