
3. Representation theory

1. 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?
A. Snowden: Probably false for $\mathrm{FI}_d$ ($d>1$).

[Ra] Eric Ramos, On the degree-wise 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?
Known question, but maybe $\mathrm{FI}$-perspective brings something new.
• 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.