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3. Representation theory

    1. Coherence

          By [Theorem B, Ra], the category of $\mathrm{FI}$-modules presented in finite degree is abelian.

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      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 $\mathrm{VIC}(\mathbb{F})$


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          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.

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              Problem 3.3.

              [N. Harman] Does this lead to an interesting analogue of $\mathrm{VI}$-modules?

                • $\mathrm{FI}$ like categories


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                  Problem 3.4.

                  [V. Reiner] Modify $\mathrm{FI}$ to use double cover of symmetric groups.

                    • Whitehouse modules


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                      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


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                          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.