The free FI-module $M(W)$ generated by an $S_m$-module

Let $W$ be an FB-module over $k$, i.e. a functor from the groupoid of finite sets and bijections to $k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ (The category FI and FI-modules over a commutative ring); thus $W(T)$ is a $k$-module for every finite set $T$, and every bijection $\beta \text{\hspace{0.17em}⁣}:T\to T^{\mathrm{\prime }}$ gives an isomorphism $W(\beta )\text{\hspace{0.17em}⁣}:W(T)\to W(T^{\mathrm{\prime }})$. Because an FB-module is defined on all finite sets, no transport choices are needed below.

The free FI-module $M(W)$ assigns to a finite set $S$ the direct sum

$M(W{)}_S:={⨁}_{T\subseteq S}W(T)$

over all subsets $T\subseteq S$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$, The direct sum of an indexed family of modules; only finitely many summands occur). For an injection $f\text{\hspace{0.17em}⁣}:S\to S^{\mathrm{\prime }}$ and a subset $T\subseteq S$, the restriction $f{\mathrm{\mid }}_T\text{\hspace{0.17em}⁣}:T\to f(T)$ is a bijection (Injection, surjection, bijection), and the transition map $f_{\ast }\text{\hspace{0.17em}⁣}:M(W{)}_S\to M(W{)}_{S^{\mathrm{\prime }}}$ is the sum of the isomorphisms $W(f{\mathrm{\mid }}_T)\text{\hspace{0.17em}⁣}:W(T)\to W(f(T))$ over all $T\subseteq S$. This is $k$-linear and functorial, since $(f^{\mathrm{\prime }}\circ f){\mathrm{\mid }}_T=f^{\mathrm{\prime }}{\mathrm{\mid }}_{f(T)}\circ f{\mathrm{\mid }}_T$; hence $M(W)$ is an FI-module over $k$.

For $S=[n]=\{0,\dots ,n-1\}$ the symmetric group $S_n$ (The finite symmetric group $S_n$, one-line notation, and cycle notation) acts by $\sigma \cdot T:=\sigma (T)$ and $W(\sigma {\mathrm{\mid }}_T)$, and this is a left action making $M(W{)}_n$ a left $k{S}_n$-module (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements). Grouping the sum by the cardinality of $T$, the subsets of cardinality $a$ form a single $S_n$-orbit with stabilizer the ordered block subgroup $S_{\{0,\dots ,a-1\}}\times S_{\{a,\dots ,n-1\}}\cong S_a\times S_{n-a}$, so the orbit summand is identified with an induced module (The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$):

$M(W{)}_n\cong {⨁}_{a=0}^n{\mathrm{Ind}}_{S_a\times S_{n-a}}^{S_n}(W_a⊠k),$

where $W_a:=W(\{0,\dots ,a-1\})$ is the value of the FB-module on an $a$-element set, the first factor acts on $W_a$ through $W$ and the second factor acts trivially, and $⊠$ is the external tensor product of modules for the two direct factors (The outer induction product of symmetric-group characters, The character ring of a direct product is the tensor product of the factor character rings, The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums).

Adjunction. Let $\pi$ be the functor from FI-modules to FB-modules that forgets the maps between distinct cardinalities. The inclusion of the $T=S$ summand is a natural map ${\eta }_W\text{\hspace{0.17em}⁣}:W\to \pi M(W)$, and evaluating a map $F\text{\hspace{0.17em}⁣}:M(W)\to V$ of FI-modules on that summand yields a map of FB-modules $\pi F\circ {\eta }_W\text{\hspace{0.17em}⁣}:W\to \pi V$. Conversely, a map of FB-modules $\phi \text{\hspace{0.17em}⁣}:W\to \pi V$ extends uniquely to a map of FI-modules $\tilde{\phi }\text{\hspace{0.17em}⁣}:M(W)\to V$ whose value on the summand $W(T)\subseteq M(W{)}_S$ is $V({\iota }_{T,S})\circ {\phi }_T$, with ${\iota }_{T,S}\text{\hspace{0.17em}⁣}:T\to S$ the inclusion; the components are compatible with injections and the two constructions are inverse, so ${\mathrm{Hom}}_{FI\text{-}Mod}(M(W),V)\cong {\mathrm{Hom}}_{FB\text{-}Mod}(W,\pi V)$ naturally in $V$ and $W$: $M$ is left adjoint to $\pi$, with unit $\eta$.

For $m\ge 0$ write $k[S_m]$ for the FB-module with $k[S_m](T)$ the free $k$-module on the bijections $[m]\to T$, and zero when $\mathrm{\mid }T\mathrm{\mid }\mathrm{\ne }m$; its value on an $m$-element set is the regular representation of $S_m$. Put $M(m):=M(k[S_m])$. Then $M(m{)}_S$ has as $k$-basis the injections $[m]\hookrightarrow S$ — equivalently, the ordered sequences of $m$ distinct elements of $S$ — and for a partition $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }=m$ and a field $k$ of characteristic $0$, put $M(\lambda ):=M(S_k^{\lambda })$, where the FB-module $S_k^{\lambda }$ is supported on $m$-element sets and $S_k^{\lambda }(T):=k[\mathrm{Iso}([m],T)]{\otimes }_{k{S}_m}{S}_k^{\lambda }$. By the adjunction, a map $M(m)\to V$ of FI-modules corresponds naturally to the choice of an element of $V_m$: the module $M(m)$ represents the functor $V\mapsto V_m$. All constructions are given by finitely many canonical bijections of finite sets; no choice principle is used.