Generalizations of the Exterior and Symmetric Power Functors on Categories of Modules using Coinvariants of Tensor Power Functors

Abstract

This work, broadly speaking, is a study of coinvariants of abstract group actions on functors. We discuss background on coinvariants of group actions on objects in categories in general, as we benefit from taking the perspective of a functor $F:C\to D$ as an object of the category of functors from $C$ to $D$. That said, the focus of this work concerns the $n$th tensor power functor $T^n$ on the category of modules over a commutative unital ring $k$. We show that the $k$-algebra of endomorphisms of $T^n$ is isomorphic to the group algebra $k S_n$. This allows us to identify specific groups of automorphisms of $T^n$ and investigate the resulting functors of coinvariants. For example, the $n$th symmetric power functor $Sym^n$ and the $n$th exterior power functor $\wedge^n$ are coinvariants of $T^n$ with respect to actions of the symmetric group $S_n$. Hence, the study of coinvariants of $T^n$ with respect to group actions is a way of generalizing these familiar functors. We give examples of groups $G_1$ and $G_2$ of automorphisms of $T^n$ over $\mathbb{Z}$ such that $|G_1|$ is countably infinite whilst $|G_2|$ is finite, and they give rise to canonically isomorphic coinvariants of the functor $T^n$. We also explore the topic of sequences of groups $G_n$ with actions on components of a graded algebra $S$. We provide sufficient conditions for when the resulting direct sum of modules of coinvariants is a quotient algebra of $S$. For example, this condition implies that for any $R$-module $M$, the actions of the sequence of alternating groups $A_n$ on the components of the tensor algebra $T(M)$ induces an algebra $C$ of coinvariants. While $C$ is distinct from the algebras $Sym(M)$ and $\wedge(M)$, we observe that for finitely generated $R$-modules $M$ and sufficiently large $n$, there are isomorphisms $C_n\cong Sym^n(M)$.