Uniform representation stability for consistent sequences of $S_n$-modules

Let $k$ be a field of characteristic $0$. A consistent sequence is a sequence $(V_n,{\phi }_n{)}_{n\ge 0}$ in which each $V_n$ is a finite-dimensional left $k{S}_n$-module and each ${\phi }_n\text{\hspace{0.17em}⁣}:V_n\to V_{n+1}$ is $S_n$-equivariant, the target being regarded as an $S_n$-module by restriction along the standard inclusion $S_n\hookrightarrow S_{n+1}$ that fixes $n$. An FI-module $V$ over $k$ with finite-dimensional values (The category FI and FI-modules over a commutative ring) determines such a sequence by $V_n:=V([n])$ and ${\phi }_n:=f_{\ast }$ for the standard inclusions $f\text{\hspace{0.17em}⁣}:[n]\to [n+1]$; these maps are $S_n$-equivariant by functoriality and the conjugation action, since $f\circ \sigma ={\mathrm{\Phi }}_f(\sigma )\circ f$ with ${\mathrm{\Phi }}_f(\sigma )$ the extension of $\sigma$ by the fixed point $n$.

The sequence is uniformly representation stable with stable range $n\ge N$ when all three of the following hold.

1. (I) Injectivity. The map ${\phi }_n\text{\hspace{0.17em}⁣}:V_n\to V_{n+1}$ is injective for every $n\ge N$.
2. (II) Surjectivity. The $S_{n+1}$-span of the image ${\phi }_n(V_n)$ is all of $V_{n+1}$ for every $n\ge N$.
3. (III) Multiplicities. For each partition $\lambda$ and each $n$ with $n\ge \mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1$, let $V(\lambda {)}_n$ be the irreducible $k{S}_n$-module indexed by the first-row padding $\lambda [n]=(n-\mathrm{\mid }\lambda \mathrm{\mid },{\lambda }_1,{\lambda }_2,\dots )$ (First-row padding of a partition), namely the scalar extension $k{\otimes }_{\mathbb{Q}}{S}_{\mathbb{Q}}^{\lambda [n]}$ of the rational Specht model, which is the unique irreducible $k{S}_n$-module with that label (Specht modules classify the irreducibles over every field of characteristic zero; over $\mathbb{C}$ this is the published complex Specht classification Complex Specht modules are irreducible). Decompose $V_n\cong {⨁}_{\lambda }{c}_{\lambda ,n}V(\lambda {)}_n$ with multiplicities $c_{\lambda ,n}\ge 0$, the sum over those $\lambda$ with $n\ge \mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1$. Then for every fixed $\lambda$ the multiplicity $c_{\lambda ,n}$ is independent of $n$ for all $n\ge N$, with $c_{\lambda ,n}:=0$ when $n<\mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1$ so that undefined padded modules do not leave undefined multiplicities.

The sequence is uniformly representation stable when it is so for some $N\ge 0$. All three conditions are part of the definition, and (II) is not a consequence of (I) and (III): take $V_n=k^{[n]}$ with its permutation action and, for $n\ge 1$, define ${\phi }_n(e_i)=e_i-e_n$; let ${\phi }_0$ be the zero map from $V_0=0$. These maps are $S_n$-equivariant and injective, since the first $n$ coordinates of ${\sum }_i{a}_i(e_i-e_n)$ are the $a_i$. For $n\ge 2$, $V_n$ is the direct sum of the trivial line and the irreducible sum-zero space $S_k^{(n-1,1)}$, so its padded multiplicities are constant. To see the latter identification directly, the $(n-1,1)$-tabloid module is the permutation module on the singleton second row, and its polytabloids are the differences $e_i-e_j$, which span the sum-zero space. Every image of ${\phi }_n$ lies in the sum-zero submodule of $V_{n+1}$, so its $S_{n+1}$-span misses the trivial quotient. Thus (I) and (III) hold for $n\ge 2$, while (II) fails. No choice principle is used.