Character Polynomials, FI-Modules, and Representation Stability

Let $FI$ be the category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) whose objects are the finite sets and whose morphisms are the injections; composition is composition of functions and identities are identity functions. Every finite set $S$ is in bijection with exactly one of the sets $[n]:=\{0,1,\dots ,n-1\}$, $n\ge 0$, and injections between these representatives form a full skeletal subcategory: its inclusion is fully faithful, and every object of $FI$ is isomorphic to one of its objects. This skeletal description requires no simultaneous selection of bijections or quasi-inverse on the class of all finite sets; under the realization $S_n=\mathrm{Sym}(\{0,\dots ,n-1\})$ of The finite symmetric group $S_n$, one-line notation, and cycle notation this gives ${\mathrm{End}}_{FI}([n])=S_n$ for every $n\ge 0$, and between different objects there are no endomorphisms.

An FI-module over a commutative ring $k$ (Commutative ring) is a covariant functor $V\text{\hspace{0.17em}⁣}:FI\to k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ (Covariant functor, identity functor, composite functor, and contravariant functor, Unital left and right modules over a ring; unqualified module means left module). Write $V_S:=V(S)$, $V_n:=V([n])$ and $f_{\ast }:=V(f)\text{\hspace{0.17em}⁣}:V_S\to V_T$ for a morphism $f\text{\hspace{0.17em}⁣}:S\to T$; functoriality means $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{i}\mathrm{d}}_S{)}_{\ast }={\mathrm{i}\mathrm{d}}_{V_S}$. Since ${\mathrm{End}}_{FI}([n])=S_n$, the maps $V_n\to V_n$ induced by permutations make $V_n$ a left $k{S}_n$-module via the group ring (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures): functoriality gives ${\sigma }_{\ast }{\tau }_{\ast }=(\sigma \tau {)}_{\ast }$ and ${(\mathrm{i}\mathrm{d})}_{\ast }=\mathrm{i}\mathrm{d}$, and the group-ring axioms follow.

For an injection $f\text{\hspace{0.17em}⁣}:[m]\to [n]$ define ${\mathrm{\Phi }}_f\text{\hspace{0.17em}⁣}:S_m\to S_n$ by ${\mathrm{\Phi }}_f(\sigma )(y):=f(\sigma (f^{-1}(y)))$ for $y\in f([m])$ and ${\mathrm{\Phi }}_f(\sigma )(y):=y$ for $y\mathrm{\notin }f([m])$; then ${\mathrm{\Phi }}_f$ is a group homomorphism, because $f^{-1}(y)$ is defined on $f([m])$, the two factors of a product commute through $f$, and both sides fix the complement pointwise. Since $f\circ \sigma ={\mathrm{\Phi }}_f(\sigma )\circ f$, functoriality gives $f_{\ast }\circ {\sigma }_{\ast }=({\mathrm{\Phi }}_f(\sigma ){)}_{\ast }\circ f_{\ast }$ for all $\sigma \in S_m$: each $f_{\ast }\text{\hspace{0.17em}⁣}:V_m\to V_n$ is equivariant for the conjugation homomorphism ${\mathrm{\Phi }}_f$ determined by $f$.

A sub-FI-module of $V$ is a subfunctor $W\subseteq V$ (Submodule of a module): submodules $W_S\le V_S$ with $f_{\ast }(W_S)\subseteq W_T$ for all injections $f\text{\hspace{0.17em}⁣}:S\to T$. A map of FI-modules $V\to W$ is a natural transformation, and kernels, images and cokernels of such maps are computed pointwise, so that e.g. $\ker (F{)}_S=\ker (F_S\text{\hspace{0.17em}⁣}:V_S\to W_S)$ with the restrictions of the transition maps (Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets, Abelian category). With these pointwise operations the category of FI-modules over $k$ inherits its additive structure and its abelian-category structure from $k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ (Modules over a ring form an abelian category): finite direct sums, kernels, cokernels and the compatibility axioms are checked objectwise, and a pointwise surjective map is an epimorphism and a pointwise injective map is a monomorphism.

An FB-module over $k$ is a functor from the groupoid $FB$ of finite sets and bijections to $k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$, with natural transformations as maps. Since all morphisms of $FB$ are isomorphisms and every finite set is isomorphic to exactly one $[n]$, an FB-module is determined up to isomorphism by the $k{S}_n$-modules $W_n$ alone and carries no transition maps between different $n$. No choice principle is used beyond the definable-class conventions of Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

Let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring) and let $\mathrm{\Sigma }$ be a set of supported elements $(S,v)$, where $S$ is a finite set and $v\in V_S$. The support $S$ is part of the generator data, even when distinct values of $V$ have overlapping underlying sets. Write ${\mathrm{\Sigma }}_S:=\{v\in V_S:(S,v)\in \mathrm{\Sigma }\}$; a sub-FI-module $W$ contains $\mathrm{\Sigma }$ when $v\in W_S$ for every $(S,v)\in \mathrm{\Sigma }$. The sub-FI-module generated by $\mathrm{\Sigma }$, written ${\mathrm{span}}_V(\mathrm{\Sigma })$, is the intersection of all sub-FI-modules of $V$ that are subfunctors and contain $\mathrm{\Sigma }$ (Submodule of a module, The one-step submodule criterion; intersections and sums of submodules are submodules for the objectwise intersection of submodules). The family is nonempty because $V$ itself qualifies, and the intersection is a sub-FI-module, since each transition map $f_{\ast }$ preserves every member and hence their intersection; thus ${\mathrm{span}}_V(\mathrm{\Sigma })$ is the smallest sub-FI-module of $V$ containing $\mathrm{\Sigma }$.

Explicitly, ${\mathrm{span}}_V(\mathrm{\Sigma }{)}_T$ is the $k$-span of all elements $f_{\ast }(v)$ with $(S,v)\in \mathrm{\Sigma }$ and $f\text{\hspace{0.17em}⁣}:S\to T$ a morphism of $FI$: the set of all finite $k$-linear combinations of such elements is a sub-FI-module of $V$ containing $\mathrm{\Sigma }$, because $g_{\ast }(f_{\ast }(v))=(g\circ f{)}_{\ast }(v)$, and every sub-FI-module containing $\mathrm{\Sigma }$ contains each $f_{\ast }(v)$; hence it equals the intersection. Call $V$ finitely generated when ${\mathrm{span}}_V(\mathrm{\Sigma })=V$ for some finite $\mathrm{\Sigma }$, and generated in degree $\le d$ when ${\mathrm{span}}_V(V_{\le d})=V$, where $V_{\le d}=\{([m],v):0\le m\le d,v\in V_{[m]}\}$ uses the finite skeleton of FI; transport by bijections shows that its span also contains every $V_S$ with $\mathrm{\mid }S\mathrm{\mid }\le d$. The generation degree of $V$ is the least $d\ge 0$ such that $V$ is generated in degree $\le d$, and $\mathrm{\infty }$ when no such $d$ exists (Generated submodule, cyclic and finitely generated modules, module basis and free module).

Finite generation of $V$ and pointwise finite generation of the individual $k$-modules $V_S$ are different conditions: the pointwise property is implied by finite generation (as shown below by Generation in bounded degree presents an FI-module as a quotient of free FI-modules) but does not imply it, and no pointwise finite generation or finite-dimensionality is built into either notion. A finite generating set gives a finite degree bound: take the maximum cardinality of its supporting sets (and $0$ for the empty generating set). When a generator is written simply as $v\in V_S$, its specified support $S$ is understood, so ${\mathrm{span}}_V(v)$ means ${\mathrm{span}}_V(\{(S,v)\})$. Generation in bounded degree alone does not imply finite generation. No choice principle is used: ${\mathrm{span}}_V(\mathrm{\Sigma })$ is defined by an intersection of subfunctors, and its explicit description involves only finite $k$-linear combinations.

A graded $k[T]$-module consists of $k$-modules $U_n$, $n\ge 0$, together with $k$-linear maps $T=T_n\text{\hspace{0.17em}⁣}:U_n\to U_{n+1}$ for all $n\ge 0$. A morphism $U\to U^{\mathrm{\prime }}$ of graded $k[T]$-modules is a family of $k$-linear maps $u_n\text{\hspace{0.17em}⁣}:U_n\to U_n^{\mathrm{\prime }}$ with $u_{n+1}\circ T_n=T_n^{\mathrm{\prime }}\circ u_n$ for all $n$.

Fix $a\ge 0$, a set $\mathbf{a}$ of cardinality $a$, and a functorial disjoint union of finite sets. Let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring) and let $n\ge 0$. Put

${\mathrm{\Phi }}_a(V{)}_n:=(V_{\mathbf{a}\bigsqcup n}{)}_{S_n},$

the coinvariants of the $k{S}_n$-module $V_{\mathbf{a}\bigsqcup n}$: the quotient by the span of the elements $\sigma \cdot v-v$, $\sigma \in S_n$, $v\in V_{\mathbf{a}\bigsqcup n}$ (The coinvariants functor, Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets). Equivalently ${\mathrm{\Phi }}_a(V{)}_n=k{\otimes }_{k{S}_n}{V}_{\mathbf{a}\bigsqcup n}$, the two descriptions being identified by the balancing argument of 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 and the corresponding identification for $\mathbb{Z}$ in The coinvariants functor; the group ring $k{S}_n$ is a unital $k$-algebra with basis the group elements (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$).

To define the transition map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$, choose an injection $f\text{\hspace{0.17em}⁣}:n\to n+1$. It induces the injection ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f\text{\hspace{0.17em}⁣}:\mathbf{a}\bigsqcup n\to \mathbf{a}\bigsqcup n+1$ and hence a $k$-linear map $(V_{\mathbf{a}\bigsqcup n}\to V_{\mathbf{a}\bigsqcup n+1})$. This map is $S_n$-equivariant, where $S_n$ acts on the target through the injection ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$, so it descends to the quotient map $T=T_f$ on coinvariants. The map $T_f$ is independent of $f$: for two injections $f,g\text{\hspace{0.17em}⁣}:n\to n+1$ the composite $g\circ f^{-1}$ is a bijection $f(n)\to g(n)$, which extends along the complement to a permutation $\tau \in S_{n+1}$ with $g=\tau \circ f$; then ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup g=({\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup \tau )\circ ({\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f)$, and the map induced by ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup \tau$ on $(V_{\mathbf{a}\bigsqcup n+1}{)}_{S_{n+1}}$ is the identity because it comes from the action of the group element $\tau$. Thus $T$ is well defined and $({\mathrm{\Phi }}_a(V),T)$ is a graded $k[T]$-module.

A map of FI-modules $\alpha \text{\hspace{0.17em}⁣}:V\to W$ gives $k$-linear maps ${\alpha }_{\mathbf{a}\bigsqcup n}\text{\hspace{0.17em}⁣}:V_{\mathbf{a}\bigsqcup n}\to W_{\mathbf{a}\bigsqcup n}$ commuting with the $S_n$-actions and with the maps induced by ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$, hence maps $({\mathrm{\Phi }}_a\alpha {)}_n$ of coinvariants commuting with $T$; so ${\mathrm{\Phi }}_a$ is a functor from FI-modules over $k$ to graded $k[T]$-modules.

Permutations of $\mathbf{a}$ commute with the moving-block action of $S_n$, so ${\mathrm{\Phi }}_a(V{)}_n$ also carries an $S_a$-action. Since ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$ commutes with these permutations, every $T$ is $S_a$-equivariant. Naturality has the same equivariance.

Exactness. Coinvariants of a group module are right exact (The coinvariants functor is right exact), so the functor $M\mapsto M_{S_n}\cong k{\otimes }_{k{S}_n}M$ is right exact and each ${\mathrm{\Phi }}_a$ is right exact in every degree. If $k$ contains $\mathbb{Q}$, then $1\mathrm{/}n!$ exists in $k$ and the averaging map $s\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to V_{\mathbf{a}\bigsqcup n}$, $s([v])=\frac{1}{n!}{\sum }_{\sigma \in S_n}\sigma \cdot v$, is a well-defined $k$-linear section of the quotient map that is natural in $V$; it identifies ${\mathrm{\Phi }}_a(V{)}_n$ with the invariants $V_{\mathbf{a}\bigsqcup n}^{S_n}$. Under this natural identification ${\mathrm{\Phi }}_a$ becomes the invariants functor, which is left exact (kernels of equivariant maps are computed pointwise), and therefore ${\mathrm{\Phi }}_a$ is exact; in that case the induced map ${\mathrm{\Phi }}_a(W)\to {\mathrm{\Phi }}_a(V)$ is injective for every sub-FI-module $W\subseteq V$ (Submodule of a module, Module homomorphism and isomorphism, kernel, image and cokernel), so ${\mathrm{\Phi }}_a(W)$ is identified with a submodule of ${\mathrm{\Phi }}_a(V)$. Finite averaging uses only the finite groups $S_n$; no choice principle is used.

Let $V$ be an FI-module over $k$ and let ${\mathrm{\Phi }}_a$ and $T$ be the coinvariant functors and transition maps of The coinvariant functors ${\mathrm{\Phi }}_a$ and the transition maps $T$. The injectivity degree $\mathrm{inj-deg}(V)$ is the least $s\ge 0$ such that for every $a\ge 0$ the transition map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$ is injective for all $n\ge s$; the surjectivity degree $\mathrm{surj-deg}(V)$ is the least $s\ge 0$ such that for every $a\ge 0$ the map $T$ is surjective for all $n\ge s$. The stability degree is

$\mathrm{stab-deg}(V):=\max (\mathrm{inj-deg}(V),\mathrm{surj-deg}(V))\mathrm{.}$

Each of the three is $+\mathrm{\infty }$ when no such $s$ exists; here the bounds are read as elements of the extended nonnegative integers $\mathbb{N}\cup \{+\mathrm{\infty }\}$ with the evident order, in which every nonempty subset of $\mathbb{N}$ has a least element (The natural numbers $\mathbb{N}$ (von Neumann)), so "the least $s$" is well defined. Equivalently, $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$ is an isomorphism for all $a\ge 0$ and all $n\ge \mathrm{stab-deg}(V)$: the map is an isomorphism exactly when it is both injective and surjective, and for finite bounds a threshold that controls all $a$ simultaneously is the larger of the two thresholds, while each of the two stated bounds remains the least threshold controlling its own property.

The three bounds are kept distinct: $\mathrm{stab-deg}(V)$ is the maximum of the other two, and nothing in the definition requires $\mathrm{inj-deg}(V)=\mathrm{surj-deg}(V)$. No finiteness is asserted: each bound is allowed to be $+\mathrm{\infty }$. No choice principle is used.

Let $F$ be a field of characteristic $0$ and let $n\ge 0$. For every partition $\lambda ⊢n$ the field-valued Specht module $S_F^{\lambda }$ of Integral and field-valued Specht modules is nonzero and irreducible, distinct partitions give non-isomorphic modules, and every irreducible $F{S}_n$-module is isomorphic to exactly one of them. Consequently, by Maschke's theorem, every finite-dimensional $F{S}_n$-module is a direct sum of copies of the $S_F^{\lambda }$; thus the irreducible $F{S}_n$-modules are indexed by the partitions of $n$ and the modules $\{S_F^{\lambda }:\lambda ⊢n\}$ form a complete irredundant list.

This is the constituent convention used throughout the pair: over $F=\mathbb{C}$ it specialises to the published complex classification, and over a general characteristic-zero field $F$ it is proved by extending the rational Specht models along $\mathbb{Q}\to F$, never by a scalar extension from $\mathbb{C}$ to $F$.

Let $k$ be a field of characteristic $0$ and let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring). For a partition $\lambda$ and $n\ge \mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1$ put $\lambda [n]:=(n-\mathrm{\mid }\lambda \mathrm{\mid },{\lambda }_1,{\lambda }_2,\dots )$, the first-row padding (First-row padding of a partition), with $\mathrm{\mid }\mathrm{\varnothing }\mathrm{\mid }={\mathrm{\varnothing }}_1=0$ and $\mathrm{\varnothing }[0]=\mathrm{\varnothing }$, and let

$V(\lambda {)}_n:=k{\otimes }_{\mathbb{Q}}{S}_{\mathbb{Q}}^{\lambda [n]}$

be the scalar extension of the rational Specht model of $\lambda [n]$ (Complex Specht modules are irreducible, Specht modules classify the irreducibles over every field of characteristic zero). By the classification lemma, $V(\lambda {)}_n$ is the irreducible $k{S}_n$-module with label $\lambda [n]$, and every irreducible $k{S}_n$-module is isomorphic to exactly one $V(\lambda {)}_n$: indeed, every partition $\mu =({\mu }_1,{\mu }_2,\dots )$ of $n$ is uniquely $\lambda [n]$ for the partition $\lambda =({\mu }_2,{\mu }_3,\dots )$ obtained by deleting the first row, and $n\ge \mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1$ holds precisely because ${\mu }_1\ge {\mu }_2$; consequently the labels $\lambda [n]$ run over the partitions of $n$ without repetition. At $n=0$ the only label and its tail are both $\mathrm{\varnothing }$, and $V(\mathrm{\varnothing }{)}_0=k$ with the trivial $S_0$-action.

Say that $V(\lambda {)}_n$ occurs as a constituent of $V_n$ when there are $k{S}_n$-submodules $U\subseteq W\subseteq V_n$ with $W\mathrm{/}U\cong V(\lambda {)}_n$ (Submodule of a module, Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets); this is a condition on subquotients and requires no decomposition or finite-dimensionality of $V_n$. The weight of $V$ is

$\mathrm{weight}(V):=\sup \{\mathrm{\mid }\lambda \mathrm{\mid }:V(\lambda {)}_n\text{ occurs as a constituent of }{V}_n\text{ for some }n\ge \mathrm{\mid }\lambda \mathrm{\mid }+{\lambda }_1\},$

the supremum over all pairs $(\lambda ,n)$ as displayed. The set of integers displayed is a set of nonnegative integers, so the supremum is well defined in $\mathbb{N}\cup \{\mathrm{\infty }\}$; it is $0$ for $V=0$, and $\mathrm{weight}(V)=\mathrm{\infty }$ exactly when the displayed set is unbounded. No finiteness of $\mathrm{weight}(V)$ is asserted.

If $W$ is a subquotient of $V$, meaning that $W\cong U^{\mathrm{\prime }}\mathrm{/}U$ for sub-FI-modules $U\subseteq U^{\mathrm{\prime }}\subseteq V$ computed pointwise, then $\mathrm{weight}(W)\le \mathrm{weight}(V)$: a constituent of $W_n$ is a subquotient of the subquotient $U_n^{\mathrm{\prime }}\mathrm{/}{U}_n$ of $V_n$, hence a subquotient of $V_n$, so it contributes the same label $\mathrm{\mid }\lambda \mathrm{\mid }$ to the supremum for $V$. No choice principle is used: the labels $\lambda [n]$, constituents and the supremum are set-theoretic data, and no bases or decompositions are chosen.

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.