
## 1. $\mathrm{FI}$-modules and tca’s

1. ### Nate Harman’s question on lifting $\mathrm{FI}$-modules from $\mathbb{C}$ to $\mathbb{Z}$

#### Problem 1.1.

[Nate Harman] Given a map $\phi_{\mathbb{Q}} \colon V_{\mathbb{Q}} \to W_{\mathbb{Q}}$ of finitely generated $\mathrm{FI}$-modules over $\mathbb{Q}$ (or may be over an algebraically closed field), and choices $V, W$ of lattices in $V_{\mathbb{Q}}$ and $W_{\mathbb{Q}}$. If $p$ is sufficiently large, is $V \otimes_{\mathbb{Z}} \mathbb{F}_p \to W \otimes_{\mathbb{Z}} \mathbb{F}_p$ independent of the choice of $V$ and $W$ up to isomorphism?
• ### Noetherianity of $\mathrm{FI}$-algebras

#### Problem 1.2.

Let $V_n = \mathbf{k}[x_1 , x_2, \ldots x_n]$. Find a concrete example of a sub-$\mathrm{FI}$-algebra which is not noetherian. Is there a version for finitely presented modules? Does it depend on characteristic?
• ### Classify torsion free injectives

#### Problem 1.3.

[John Wiltshire-Gordon] What’s the injective hull of the trivial $\mathrm{FI}$-modules ($V_n = \mathrm{triv}_n$ for each $n$)? In general, classify all indecomposable injectives. Is it true that these indecomposable injectives are degree-wise finitely generated (same question over a more general combinatorial category).
• ### Highest weight category

#### Problem 1.4.

[R. Nagpal] Working over a field. Is the Serre quotient $\mathrm{Mod}_{\mathrm{FI}}/\mathrm{Mod}_{\mathrm{FI}}^{\mathrm{tors}}$ a highest weight category?
• ### Noetherianity of degree two tca’s

#### Problem 1.5.

1. {T. Church} Is $\mathrm{Sym}(\mathrm{Sym}^2)$ noetherian over $\mathbb{Z}$ $($or over a general noetherian ring$)$?
2. {J. Wilson} Same question for $\bigwedge(\mathrm{Sym}^2)$.
• ### Bounds on syzygies of modules over degree two tca’s

#### Problem 1.6.

[J. Miller] Let $A = \mathrm{Sym}(\mathrm{Sym}^2)$ or $\bigwedge(\mathrm{Sym}^2)$, and let $M$ be an $A$-module generated in degree $d$ and related in degree $r$. Are there any good bounds on the syzygies? In other words, bound generators of the terms of the free resolutions. Can start with characteristic $0$.
There are no linear bounds, that is, regularity is infinite. As an example, take $A$ modulo a determinantal ideal (see [section 6.3, W]). The calculations have been done in characteristic zero only though.

[W] Jerzy Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge University Press, Cambridge, 2003.
• ### Hilbert series of modules over degree two tca’s

#### Problem 1.7.

[A. Snowden] What can we say about Hilbert series of modules over $A = \mathrm{Sym}(\mathrm{Sym}^2)$. More precisely, take $M$ to be $A$ modulo a determinantal ideal. Is it true that its $($non-exponential$)$ Hilbert series is algebraic? Assume characteristic $0$.
Concretely, let $\mathrm{H}_m(t) = \sum_{|\lambda| = n} \dim(M_{2\lambda}) t^{2n}.$ Is $\mathrm{H}_m(t)$ algebraic?

Note that $\mathrm{H}_m(t)$ is the ordinary Hilbert series of $A$ modulo the $m$th determinantal ideal. If $m = \infty$, then $\mathrm{H}_m(t)$ is not algebraic. $H_2(t)$ is algebraic and is closely related to Catalan numbers.
• ### $q$-tca’s

Berenstein–Zwicknagl define quantum analog of $\mathrm{Sym}(\mathbf{S}_{\lambda})$ in [BZ]. For small $\lambda$, this behaves like the classical case, but for large $\lambda$ (and $q$ generic) its isotypic decomposition has smaller multiplicities than the classical case.

#### Problem 1.8.

[D. Barter] Are modules over this algebra $($as $q$-tca’s$)$ equivalent to modules in the classical case $($as tca’s$)$?
$\mathrm{Sym}(\mathbf{S}_1 \oplus \mathbf{S}_1)$ already smaller than classical case, question is interesting here. (Already known for $\mathrm{Sym}(\mathbf{S}_1)$.)

[BZ] Arkady Berenstein, Sebastian Zwicknagl, Braided symmetric and exterior algebras, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3429–3472

Cite this as: AimPL: Representation stability, available at http://aimpl.org/repnstability.