Free FI-modules have computable stability degree and weight

The $S_n$-orbits of the subset summands of $M(W{)}_{\mathbf{a}\bigsqcup [n]}$ are indexed by $(U,r)$, where $U\subseteq \mathbf{a}$ and $0\le r\le n$: take the representative $U\bigsqcup [r]$ with $[r]$ in the moving block. Its stabilizer is $S_r\times S_{n-r}$, and the second factor acts trivially on $W(U\bigsqcup [r])$. Transporting the orbit summands to the representative identifies their coinvariants with $(W(U\bigsqcup [r]){)}_{S_r}$: different transports differ by a stabilizer element, so they give the same map after quotienting, and the representative inclusion is inverse. Thus ${\mathrm{\Phi }}_{a}M(W{)}_n\cong {⨁}_{U\subseteq \mathbf{a}}{⨁}_{r=0}^n(W(U\bigsqcup [r]){)}_{S_r}$. Choosing a single bijection $\mathbf{b}\to U$, with $b=\mathrm{\mid }U\mathrm{\mid }$, identifies this factor with ${\mathrm{\Phi }}_b(W{)}_r:=(W(\mathbf{b}\bigsqcup [r]){)}_{S_r}$; for an FB-module this denotes only a degreewise quotient, without transition maps. The standard inclusion $[n]\hookrightarrow [n+1]$ fixes every representative with $r\le n$, so $T$ acts as the identity on every existing factor and only adds factors with $r=n+1$.

Let $k$ have characteristic $0$ and let $\mathrm{\mid }\lambda \mathrm{\mid }=m$. For $n\ge m$ the identification $M(S_k^{\lambda }{)}_n\cong {\mathrm{Ind}}_{S_m\times S_{n-m}}^{S_n}(S_k^{\lambda }⊠k)$ of [F1] together with the branching rule [F3] shows that the constituents of $M(\lambda {)}_n$ are the Specht modules $S^{\nu }$ over the partitions $\nu ⊢n$ obtained from $\lambda$ by adding one box in each of $n-m$ different columns; for $n<m$ the value is $0$. For such a $\nu$, the added boxes occupy $n-m$ distinct columns, so ${\nu }_1\ge n-m$; writing $\nu =\eta [n]$ this gives $\mathrm{\mid }\eta \mathrm{\mid }=n-{\nu }_1\le m$. Hence every constituent $V(\eta {)}_n$ of any $M(\lambda {)}_n$ has $\mathrm{\mid }\eta \mathrm{\mid }\le m$, so $\mathrm{weight}(M(\lambda ))\le m$ by [F5].

By step 1.1 the map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_{a}M(W{)}_n\to {\mathrm{\Phi }}_{a}M(W{)}_{n+1}$ is injective for all $a,n$, since it is the identity on every direct summand and maps distinct summands to distinct summands; hence $\mathrm{inj-deg}(M(W))=0$. If $W_i=0$ for $i>m$, then ${\mathrm{\Phi }}_{\mathrm{\mid }U\mathrm{\mid }}(W{)}_r=0$ for $r>m$, so for $n\ge m$ the direct-sum decomposition of step 1.1 is independent of $n$ and $T$ is an isomorphism; hence $\mathrm{surj-deg}(M(W))\le m$. For $M(m)=M(k[S_m])$ the factor ${\mathrm{\Phi }}_0(k[S_m]{)}_m=(k[S_m]{)}_{S_m}\cong k$ is nonzero when $k\mathrm{\ne }0$, and it occurs in ${\mathrm{\Phi }}_0(M(m){)}_n$ only when $n\ge m$; so for $m>0$ the map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_0(M(m){)}_{m-1}\to {\mathrm{\Phi }}_0(M(m){)}_m$ is not surjective, giving $\mathrm{surj-deg}(M(m))\ge m$ and hence equality, and $\mathrm{stab-deg}(M(m))=m$. For $m=0$ the upper bound already gives equality $0$; no negative degree is used. Over the zero ring all modules vanish, so all these degrees are $0$ instead.

Conversely, for $n\ge m+{\lambda }_1$ the padded partition $\lambda [n]=(n-m,{\lambda }_1,{\lambda }_2,\dots )$ is obtained from $\lambda$ by adding one box in each of the first $n-m$ columns, so $V(\lambda {)}_n$ occurs in $M(\lambda {)}_n$ by step 1.2; therefore $\mathrm{weight}(M(\lambda ))\ge m$, and with step 1.2, $\mathrm{weight}(M(\lambda ))=m$.

For the stability degree, fix $a\ge 0$ and use the decomposition of step 1.1 for $W=S_k^{\lambda }$: the factor indexed by $U\subseteq \mathbf{a}$ is ${\mathrm{\Phi }}_{\mathrm{\mid }U\mathrm{\mid }}(S_k^{\lambda }{)}_r=(S_k^{\lambda }{)}_{S_r}$ when $\mathrm{\mid }U\mathrm{\mid }+r=m$ (and $0$ otherwise), where $S_r$ acts on the last $r$ points. By the branching rule [F3], $(S_k^{\lambda }{)}_{S_r}$ is the sum of the Specht modules obtained from $\lambda$ by removing one box from each of $r$ different columns, and by the strip criterion [F4] such a removal exists exactly when $r\le {\lambda }_1$; hence $(S_k^{\lambda }{)}_{S_r}=0$ for $r>{\lambda }_1$ and is nonzero for $r={\lambda }_1$ provided $m\ge {\lambda }_1$. Consequently, for every $n\ge {\lambda }_1$ all nonzero factors of ${\mathrm{\Phi }}_{a}M(\lambda {)}_n$ already occur in ${\mathrm{\Phi }}_{a}M(\lambda {)}_{{\lambda }_1}$, and $T$ acts as the identity on them, so $T$ is surjective for all $n\ge {\lambda }_1$; thus $\mathrm{surj-deg}(M(\lambda ))\le {\lambda }_1$.

Sharpness: if ${\lambda }_1>0$, take $a=m-{\lambda }_1\ge 0$. The factor with $U=\mathbf{a}$ and $r={\lambda }_1$ equals $(S_k^{\lambda }{)}_{S_{{\lambda }_1}}\mathrm{\ne }0$ and occurs in ${\mathrm{\Phi }}_{a}M(\lambda {)}_n$ exactly when $n\ge {\lambda }_1$; in particular $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_{a}M(\lambda {)}_{{\lambda }_1-1}\to {\mathrm{\Phi }}_{a}M(\lambda {)}_{{\lambda }_1}$ has a nonzero target summand not met by $T$, so it is not surjective and $\mathrm{surj-deg}(M(\lambda ))\ge {\lambda }_1$. Hence $\mathrm{surj-deg}(M(\lambda ))={\lambda }_1$, and by step 2.1, which gives injectivity degree $0$, $\mathrm{stab-deg}(M(\lambda ))={\lambda }_1$. For $\lambda =\mathrm{\varnothing }$ we have $m={\lambda }_1=0$, $M(\mathrm{\varnothing })=M(k)$ has all transition maps isomorphisms and stability degree $0$.

Step 2.1 gives the general injectivity and surjectivity bounds and the exact stability degree of $M(m)$; steps 1.2 and 2.2 give $\mathrm{weight}(M(\lambda ))=\mathrm{\mid }\lambda \mathrm{\mid }$; steps 2.3 and 3.1 give $\mathrm{stab-deg}(M(\lambda ))={\lambda }_1$.