Each padded irreducible family is realised by a finitely generated FI-module

For $n\ge d$ the padded partition $\lambda [n]=(n-m,{\lambda }_1,{\lambda }_2,\dots )$ is obtained from $\lambda =({\lambda }_1,{\lambda }_2,\dots )$ by adding exactly one box in each of the first $n-m$ columns and none elsewhere, because $n-m\ge {\lambda }_1$; hence $V_{\lambda [n]}$ occurs in $M(\lambda {)}_n$ with multiplicity one by [F1] and [F2]. For $n\ge d$ let $V(\lambda {)}_n\subseteq M(\lambda {)}_n$ be the sum of all $k{S}_n$-submodules isomorphic to $V_{\lambda [n]}$, and set $V(\lambda {)}_n:=0$ for $n<d$. As the multiplicity is at most one, this sum is the unique subspace of $M(\lambda {)}_n$ isomorphic to $V_{\lambda [n]}$, so $V(\lambda {)}_n\cong V_{\lambda [n]}$ for $n\ge d$.

We show that these subspaces are preserved by transition maps. Let $f\text{\hspace{0.17em}⁣}:[p]\to [q]$ be an injection with $p\ge d$ (if $p<d$, then $V(\lambda {)}_p=0$ and there is nothing to show), so that $q\ge d$; put $Y:=f_{\ast }(V(\lambda {)}_p)\subseteq M(\lambda {)}_q$. Since $f_{\ast }$ is injective by [F1], $Y\cong V_{\lambda [p]}$. The subspace $Y$ is stable under $H:={\mathrm{\Phi }}_f(S_p)$, because $f_{\ast }\circ {\sigma }_{\ast }={\mathrm{\Phi }}_f(\sigma {)}_{\ast }\circ f_{\ast }$ for all $\sigma \in S_p$, and it is fixed pointwise by $K:=\{\tau \in S_q:\tau {\mathrm{\mid }}_{f([p])}=\mathrm{i}\mathrm{d}\}\cong S_{q-p}$, because $\tau \circ f=f$ gives ${\tau }_{\ast }\circ f_{\ast }=f_{\ast }$ by functoriality. The subgroups $H$ and $K$ commute, $H\times K$ is conjugate in $S_q$ to the standard ordered block subgroup $S_p\times S_{q-p}$, and the pair $(Y,H\text{-action})$ is isomorphic to $(V_{\lambda [p]},S_p\text{-action})$; hence the $S_q$-span $Z$ of $Y$, which is the image of the $S_q$-module map ${\mathrm{Ind}}_{H\times K}^{S_q}(Y⊠k)\to M(\lambda {)}_q$, $(u,y)\mapsto u_{\ast }y$, is a quotient of ${\mathrm{Ind}}_{S_p\times S_{q-p}}^{S_q}(V_{\lambda [p]}⊠k)$.

Every constituent $V_{\nu }$ of $Z$ occurs both in ${\mathrm{Ind}}_{S_p\times S_{q-p}}^{S_q}(V_{\lambda [p]}⊠k)$ and in $M(\lambda {)}_q\cong {\mathrm{Ind}}_{S_m\times S_{q-m}}^{S_q}(S_k^{\lambda }⊠k)$ by step 1.2; by [F2] this means that $\nu \supseteq \lambda [p]$ with ${\nu }_{i+1}\le \lambda [p{]}_i$ and $\nu \supseteq \lambda$ with ${\nu }_{i+1}\le {\lambda }_i$. Since $\lambda [p{]}_{i+1}={\lambda }_i$ for $i\ge 1$, the inequalities ${\lambda }_i=\lambda [p{]}_{i+1}\le {\nu }_{i+1}\le {\lambda }_i$ force ${\nu }_{i+1}={\lambda }_i$ for every $i\ge 1$, and $\nu ⊢q$ gives ${\nu }_1=q-{\sum }_{i\ge 1}{\lambda }_i=q-m$; hence $\nu =\lambda [q]$ is the only possible constituent of $Z$. The multiplicity of $V_{\lambda [q]}$ in $M(\lambda {)}_q$ is at most one by [F2], so every submodule of $M(\lambda {)}_q$ all of whose constituents are isomorphic to $V_{\lambda [q]}$ is contained in the isotypic piece $V(\lambda {)}_q$; in particular $Z\subseteq V(\lambda {)}_q$, hence $Y\subseteq V(\lambda {)}_q$.

For a finite set $S$ with $\mathrm{\mid }S\mathrm{\mid }=n\ge d$ define $V(\lambda {)}_S:={\beta }_{\ast }^{-1}(V(\lambda {)}_n)\subseteq M(\lambda {)}_S$ for any bijection $\beta \text{\hspace{0.17em}⁣}:S\to [n]$; this does not depend on $\beta$, since any other bijection is $\sigma \circ \beta$ with $\sigma \in S_n$ and ${\sigma }_{\ast }(V(\lambda {)}_n)=V(\lambda {)}_n$. For $\mathrm{\mid }S\mathrm{\mid }<d$ set $V(\lambda {)}_S:=0$. If $f\text{\hspace{0.17em}⁣}:S\to T$ is an injection with $\mathrm{\mid }S\mathrm{\mid }=p\ge d$ and $\alpha \text{\hspace{0.17em}⁣}:[p]\to S$, $\beta \text{\hspace{0.17em}⁣}:[q]\to T$ are bijections, write $f=\beta \circ f_0\circ {\alpha }^{-1}$ with $f_0\text{\hspace{0.17em}⁣}:[p]\to [q]$; then $f_{\ast }(V(\lambda {)}_S)={\beta }_{\ast }(f_0{)}_{\ast }{\alpha }_{\ast }^{-1}{\alpha }_{\ast }(V(\lambda {)}_p)={\beta }_{\ast }(f_0{)}_{\ast }(V(\lambda {)}_p)\subseteq {\beta }_{\ast }(V(\lambda {)}_q)=V(\lambda {)}_T$ by step 1.2, using the functoriality of $M(\lambda )$. Hence $V(\lambda )$ is a sub-FI-module of $M(\lambda )$ with the stated values.

For $n\ge d$ and any injection $f\text{\hspace{0.17em}⁣}:[d]\to [n]$, the map $f_{\ast }\text{\hspace{0.17em}⁣}:M(\lambda {)}_d\to M(\lambda {)}_n$ is injective by [F1], so its restriction $f_{\ast }(V(\lambda {)}_d)$ is nonzero because $V(\lambda {)}_d\cong V_{\lambda [d]}\mathrm{\ne }0$ [F3], and it lies in $V(\lambda {)}_n$ by step 1.2. Every injection $[d]\to [n]$ equals $\sigma \circ f$ for some $\sigma \in S_n$, and $(\sigma \circ f{)}_{\ast }={\sigma }_{\ast }\circ f_{\ast }$, so ${\mathrm{span}}_{V(\lambda )}(V(\lambda {)}_d{)}_n$ is the $S_n$-span of $f_{\ast }(V(\lambda {)}_d)$: a nonzero $k{S}_n$-submodule of the irreducible $V(\lambda {)}_n$ [F3], hence all of $V(\lambda {)}_n$. For $n<d$ both sides are $0$, so ${\mathrm{span}}_{V(\lambda )}(V(\lambda {)}_{\le d})=V(\lambda )$; thus $V(\lambda )$ is generated in degree $\le d$ by [F7], and as $V(\lambda {)}_n=0$ for $n<d$ with $V(\lambda {)}_d\mathrm{\ne }0$, its generation degree is exactly $d$. A finite $k$-basis of the finite-dimensional $V(\lambda {)}_d$ generates all of it, so this FI-module is finitely generated.

For $n\ge d$ the value $V(\lambda {)}_n\cong V_{\lambda [n]}$ is irreducible of label $\lambda [n]$, and the tail label has $\mathrm{\mid }\lambda \mathrm{\mid }=n-(n-m)=m$; a subquotient of an irreducible module is $0$ or the module itself, so $V(\lambda {)}_n$ is the only constituent of itself, its label being $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }=m$ (Weight of an FI-module over a field of characteristic zero). For $n<d$ the value is $0$. Hence $\mathrm{weight}(V(\lambda ))=m$, the supremum being attained in every degree $n\ge d$.

By step 3.1 the module $V(\lambda )$ is a sub-FI-module of $M(\lambda )$. For all $a,n$ the inclusion $V(\lambda {)}_{\mathbf{a}\bigsqcup n}\subseteq M(\lambda {)}_{\mathbf{a}\bigsqcup n}$ induces an injective map ${\mathrm{\Phi }}_{a}V(\lambda {)}_n\to {\mathrm{\Phi }}_{a}M(\lambda {)}_n$ because $k$ contains $\mathbb{Q}$ [F4], and by naturality of $T$ these maps commute with the transition maps. If $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_{a}M(\lambda {)}_n\to {\mathrm{\Phi }}_{a}M(\lambda {)}_{n+1}$ is injective, then $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_{a}V(\lambda {)}_n\to {\mathrm{\Phi }}_{a}V(\lambda {)}_{n+1}$ is injective as well; since $\mathrm{inj-deg}(M(\lambda ))=0$ by [F6], the latter holds for all $n\ge 0$, so $\mathrm{inj-deg}(V(\lambda ))=0$.

Let $\mu :=\lambda [d]⊢d$, so that ${\mu }_1=d-m={\lambda }_1$. By step 4.1 the module $V(\lambda )$ is generated in degree $\le d$, so the evaluation map $e\text{\hspace{0.17em}⁣}:M(V(\lambda {)}_d)\to V(\lambda )$ is surjective by [F7]; since $V(\lambda {)}_d\cong V_{\mu }$ is irreducible, $M(V(\lambda {)}_d)\cong M(\mu )$ by the additivity of $M$ in [F1]. For any surjection $q\text{\hspace{0.17em}⁣}:U\twoheadrightarrow W$ of FI-modules and all $a,n$, the map ${\mathrm{\Phi }}_a(q{)}_n$ is surjective because ${\mathrm{\Phi }}_a$ preserves surjections [F4], and naturality gives $T_W\circ {\mathrm{\Phi }}_a(q{)}_n={\mathrm{\Phi }}_a(q{)}_{n+1}\circ T_U$; hence $T_U$ surjective implies $T_W$ surjective, so $\mathrm{surj-deg}(W)\le \mathrm{surj-deg}(U)$ (Lemma 3.1.6 of the source). Applying this to $e$ and using [F6], $\mathrm{surj-deg}(V(\lambda ))\le \mathrm{surj-deg}(M(\mu ))={\mu }_1={\lambda }_1$.

Suppose ${\lambda }_1\ge 1$. With $a:=m$ one has $a+{\lambda }_1=d$, so by [F5] applied with $n:=d$, $a:=m$ and $\mu :=\lambda [d]$, whose first part is ${\mu }_1=d-m={\lambda }_1$, the coinvariants $(V(\lambda {)}_d{)}_{S_{{\lambda }_1}}$ are isomorphic to $V_{\lambda }\mathrm{\ne }0$, and by [F4] they are ${\mathrm{\Phi }}_{m}V(\lambda {)}_{{\lambda }_1}$ (under the identification of the set $\mathbf{m}\bigsqcup {\lambda }_1$ of size $d$ with $[d]$, which is immaterial because conjugate subgroups have isomorphic coinvariants); moreover ${\mathrm{\Phi }}_{m}V(\lambda {)}_{{\lambda }_1-1}=(V(\lambda {)}_{d-1}{)}_{S_{{\lambda }_1-1}}=0$, since $V(\lambda )$ vanishes below $d$ by step 1.1. Hence $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_{m}V(\lambda {)}_{{\lambda }_1-1}\to {\mathrm{\Phi }}_{m}V(\lambda {)}_{{\lambda }_1}$ has zero source and nonzero target, so it is not surjective and $\mathrm{surj-deg}(V(\lambda ))\ge {\lambda }_1$; with step 5.1, $\mathrm{surj-deg}(V(\lambda ))={\lambda }_1$, and with step 4.3, $\mathrm{stab-deg}(V(\lambda ))=\max (0,{\lambda }_1)={\lambda }_1$. If ${\lambda }_1=0$, then $\lambda =\mathrm{\varnothing }$, $m=d=0$ and $M(\mathrm{\varnothing })=M(k)$ has all values $k$ and all transition maps isomorphisms by [F1], so $V(\mathrm{\varnothing })=M(\mathrm{\varnothing })$ and $\mathrm{inj-deg}=\mathrm{surj-deg}=\mathrm{stab-deg}=0={\lambda }_1$.

Steps 1.1, 1.2, 2.1 and 3.1 construct the sub-FI-module $V(\lambda )\subseteq M(\lambda )$ with values $V_{\lambda [n]}$ for $n\ge d$ and $0$ otherwise; step 4.1 shows that its generation degree is exactly $d$; step 4.2 gives $\mathrm{weight}(V(\lambda ))=m$; and steps 4.3–6.1 give $\mathrm{inj-deg}(V(\lambda ))=0$ and $\mathrm{surj-deg}(V(\lambda ))=\mathrm{stab-deg}(V(\lambda ))={\lambda }_1$. The only selections made are bijections between finite sets, and the value at each finite set was shown to be independent of the selection, so no choice principle is used.