3. Representation theory

Coherence
By [Theorem B, Ra], the category of $\mathrm{FI}$modules presented in finite degree is abelian.Problem 3.1.
[P. Patzt] Is this true for other categories?
[Ra] Eric Ramos, On the degreewise coherence of $\mathrm{FI}_G$modules 
Enrichments of $\mathrm{VIC}(\mathbb{F})$
Problem 3.2.
[K. Casto] Look at enrichments of $\mathrm{VIC}(\mathbb{F})$ $(\mathbb{F}$ a field$)$ over schemes $($i.e., use algebraic representations of $\mathbf{GL})$. What can you say about their representations? 
$\mathrm{VI}$ like categories
$\mathrm{VI}$ is obtained from $\mathrm{FI}$ by replacing finite sets with finite vector spaces. In combinatorics, often better to replace finite sets by finite projective spaces.Problem 3.3.
[N. Harman] Does this lead to an interesting analogue of $\mathrm{VI}$modules? 
$\mathrm{FI}$ like categories
Problem 3.4.
[V. Reiner] Modify $\mathrm{FI}$ to use double cover of symmetric groups. 
Whitehouse modules
Problem 3.5.
[P. Hersh] V. Reiner described Whitehouse modules as virtual representations. Is it clear why the negative part is a submodule of the positive part, i.e., can we fine explicit embedding? Is there a way to think about this in terms of $\mathrm{FI}$modules? 
Stable decomposition into irreducibles
Problem 3.6.
[N. Harman] Hard problem in modular $S_n$ representation theory: determining simple factors of Specht modules. Are stable values (for example if the representations are coming from a finitely generated $\mathrm{FI}$module) easier?
Cite this as: AimPL: Representation stability, available at http://aimpl.org/repnstability.