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For each $E$, a $G$-representation over $k$ (a $k[G]$-module), one can assign a $k[G]$-module structure to its dual as a vector space over $k$. This could be viewed as a contravariant functor $D : k[G]\textrm{-}\mathsf{Mod} \to k[G]\textrm{-}\mathsf{Mod}$ by $E \mapsto k\textrm{-}\mathsf{Vect}(k,E)$. However, this description is not complete as it misses the $k[G]\textrm{-}\mathsf{Mod}$ structure on the dual space. Is there an alternative way to describe this assignment so that the $k[G]\textrm{-}\mathsf{Mod}$ structure can be endowed 'functorially'?

I suspect that this could be done via manipulating $E$ since the $k[G]\textrm{-}\mathsf{Mod}$ stucture on $DE$ is directly associated to that on $E$. Does there exist a functor $F : k[G]\textrm{-}\mathsf{Mod} \to k[G]\textrm{-}\mathsf{Mod}$ such that $D \cong k[G]\textrm{-}\mathsf{Mod} (F-,k)$ where $k$ is endowed with the trivial action?

This is not a question on what the structure is. This is a question on how the structure can be depicted categorically. :(

Edit:

I suspect that contragredient is not a functorial construction.

SalutaFungo
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  • The most natural construction takes a left $kG$-module $M$, and the dual $\mathrm{Hom}_k(M,k)$ is a right $kG$-module. It then has to be turned into a left $kG$-module by taking the inverse. It is a special fact about groups that allows you to do this switching, so whatever description you want of the dual map, it needs to not apply to other module categories in general. – David A. Craven Jan 30 '24 at 10:09
  • Usually you would equip the category with this data directly. A standard way to do this is to treat $k[G]$ as a Hopf algebra, with the algebra structure as usual, the bialgebra structure having counit $1 \mapsto 1_G$ and $g \mapsto g \otimes g$ for $g \in G$, and antipode $g \mapsto g^{-1}$. The category of finite-dimensional representations of a Hopf algebra becomes a monoidal category with duals in a standard way. – Joppy Feb 02 '24 at 11:21

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$\newcommand{\V}{\mathscr{V}}\newcommand{\C}{\mathsf{C}}\newcommand{\B}{\mathbf{B}}\newcommand{\M}{\mathcal{M}}\newcommand{\vect}{\mathsf{Vect}_k}\newcommand{\op}{{^{\mathsf{op}}}}$This is kind of an enriched cotensor construction. Let $\V$ be any complete, closed symmetric monoidal category; there is a notion of $\V$-enriched category, and a notion of $\V$-module and $\V$-comodule which are special kinds of $\V$-category with associated tensor or cotensor functors (which can be promoted to $\V$-functors in an excellent way but the details are not easy to check, and seem to hardly ever be checked in fact). Sweeping definitions and details under the rug, suppose $\underline{\M}$ is a $\V$-comodule with cotensor $\pitchfork$. Then for any small $\V$-category $\underline{\C}$ there is a natural to consider and easy to write down associated $\V$-functor (there are various ways to make the cotensor act on functors, this is just one of them): $$\pitchfork:\underline{\V^{\underline{\C\op}}\boxtimes\M}\to\underline{\M^{\underline{\C}}}$$Given as the adjunct of:$$\underline{(\V^{\underline{\C\op}}\boxtimes\M)\boxtimes\C}\cong\underline{((\V\op)^{\underline{\C}}\boxtimes\C)\boxtimes\M}\overset{\mathsf{eval}\boxtimes1}{\longrightarrow}\underline{\V\op\boxtimes\M}\overset{\pitchfork}{\longrightarrow}\M$$

In the special case of $\underline{\V}=\underline{\vect}=\underline{\M}$ and $\underline{\C}=\underline{\B kG}$ the one-object $\V$-category with hom space $k[G]$, there is an isomorphism $\underline{\C}\cong\underline{\C\op}$ and your dual construction is the following $\V$-functor: $$()^\star:\underline{\V^\underline{\C}}\cong\underline{\V^{\underline{\C\op}}}\overset{(1,k)}{\longrightarrow}\underline{\V^{\underline{\C\op}}\boxtimes\V}\overset{\pitchfork}{\longrightarrow}\underline{\V^{\underline{\C}}}$$Where we use the canonical cotensoring of $\V$ over itself. In particular, the $G$-module structure comes for free.

After all, the $\vect$-category of $k$-representations of $G$ is nothing but $\underline{\vect^{\underline{\B kG}}}$ amirite? Unwinding definitions, the above could be more simply (but less generally) expressed as:

$()^\ast:\underline{\vect^{\underline{\B kG}}}\to\underline{\vect^{\underline{\B kG}}}$ is the $\vect$-functor on $G$-representations adjunct to: $$\underline{\vect^{\underline{\B kG}}\boxtimes\B kG}\overset{\text{group duality}}{\cong}\underline{(\vect\op)^{\underline{\B kG}}\boxtimes\B kG}\overset{\mathsf{eval}}{\longrightarrow}\underline{\vect\op}\overset{\underline{\vect}(-,k)}{\longrightarrow}\underline{\vect}$$

This formalism allows you to intrinsically describe the $G$-representation structure on the dual of $E$ (without making any choices, we have a map into $G$-representations) and moreover know for free that this dualisation is functorial and moreover functorial in a $k$-linear way.

FShrike
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