Bounded generation degree bounds surjectivity degree and weight

For (a), let $q\text{\hspace{0.17em}⁣}:V\twoheadrightarrow W$ be a quotient. By [F1] each $q_n$ is surjective on ${\mathrm{\Phi }}_a$, and naturality gives $T_{{\mathrm{\Phi }}_a(W)}\circ q_n=q_{n+1}\circ T_{{\mathrm{\Phi }}_a(V)}$. If $T_{{\mathrm{\Phi }}_a(V)}$ is surjective, then $T_{{\mathrm{\Phi }}_a(W)}$ is surjective as well, because every element of ${\mathrm{\Phi }}_a(W{)}_{n+1}$ is the image under the surjection $q_{n+1}$ of an element hit by $T_{{\mathrm{\Phi }}_a(V)}$. Hence the threshold controlling surjectivity of $T$ for $W$ is at most that for $V$, i.e. $\mathrm{surj-deg}(W)\le \mathrm{surj-deg}(V)$.

Let $k$ have characteristic $0$ and let $W$ be a finite-dimensional $k{S}_m$-module regarded as an FB-module supported on cardinality $m$. By [F5] $W\cong {⨁}_{\lambda }{c}_{\lambda }{S}_k^{\lambda }$ (finite sum), and by the additivity in [F3] $M(W)\cong {⨁}_{\lambda }M(S_k^{\lambda }{)}^{\oplus c_{\lambda }}$; hence $\mathrm{weight}(M(W))=\max (\{0\}\cup \{\mathrm{\mid }\lambda \mathrm{\mid }:c_{\lambda }\mathrm{\ne }0\})\le m$ by [F4] and [F6].

Let $W$ be an arbitrary $k{S}_m$-module (characteristic zero) and let $V(\eta {)}_N$ be a constituent of $M(W{)}_N$. Let $P\mathrm{/}Q\subseteq M(W{)}_N$ be a subquotient isomorphic to $V(\eta {)}_N$ and choose $x\in P\setminus Q$. The $S_N$-orbit span $X$ of $x$ is a finite-dimensional $S_N$-submodule of $M(W{)}_N$, and $X\mathrm{/}(X\cap Q)$ is a nonzero submodule of the simple $V(\eta {)}_N$, hence is isomorphic to $V(\eta {)}_N$; in particular $V(\eta {)}_N$ is a constituent of $X$. Now $X$ lies in the direct sum ${⨁}_{T\subseteq [N]}W(T)$ of [F3] over finitely many $T$, and for each of those $T$ the projection of $X$ to $W(T)$ is finite-dimensional; let $W^{\mathrm{\prime }}$ be the $k{S}_m$-submodule of $W$ generated by the finitely many images of these projections under bijections $[m]\to T$ chosen for the finitely many $m$-subsets $T$. Then $W^{\mathrm{\prime }}$ is finite-dimensional and $X\subseteq M(W^{\mathrm{\prime }}{)}_N$, so $V(\eta {)}_N$ is a constituent of $M(W^{\mathrm{\prime }}{)}_N$; by step 1.2 and [F6], $\mathrm{\mid }\eta \mathrm{\mid }\le m$. Therefore $\mathrm{weight}(M(W))\le m$ for every $k{S}_m$-module $W$.

For (b), suppose $V$ is generated in degree $\le d$. By [F3] the evaluation map ${⨁}_{m\le d}M(V_m)\twoheadrightarrow V$ is surjective, and each $V_m$ is supported in degree $m\le d$, so $\mathrm{surj-deg}(M(V_m))\le m\le d$ by [F4]; by [F2] the finite direct sum has surjectivity degree at most $d$, and step 1.1 applied to the quotient gives $\mathrm{surj-deg}(V)\le d$.

For (c), the same surjection ${⨁}_{m\le d}M(V_m)\twoheadrightarrow V$ and [F6] give $\mathrm{weight}(V)\le \mathrm{weight}({⨁}_{m\le d}M(V_m))$. A simple subquotient of a finite direct sum occurs in one summand: lift a nonzero vector of that quotient and take its finite group-orbit span. The component projections lie in finite-dimensional invariant subspaces of the summands; their finite direct sum is semisimple by [F5], so the simple quotient is isomorphic to a simple summand occurring in one component. Thus the weight of the finite direct sum is ${\max }_{m\le d}\mathrm{weight}(M(V_m))\le d$ by step 2.1. Hence $\mathrm{weight}(V)\le d$.

Step 1.1 proves (a); step 2.2 proves (b); step 3.1 proves (c).