Character polynomials in the cycle counts $X_i$

For $i\ge 1$ and $n\ge 0$ let $X_i\text{\hspace{0.17em}⁣}:S_n\to \mathbb{N}$ be the function sending a permutation $\sigma \in S_n$ to the number $X_i(\sigma ):=c_i(\sigma )$ of $i$-element orbits in its cycle type (Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type, The finite symmetric group $S_n$, one-line notation, and cycle notation, The natural numbers $\mathbb{N}$ (von Neumann)); thus $X_1(\sigma )$ is the number of fixed points of $\sigma$, and $X_i(\sigma )=0$ whenever $i>n$. Each $X_i$ is a class function on $S_n$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$): if $\tau \sigma {\tau }^{-1}$ is a conjugate of $\sigma$, then $O\mapsto \tau (O)$ is a bijection from the orbits of $\sigma$ onto the orbits of $\tau \sigma {\tau }^{-1}$ that preserves cardinalities, so $c_i(\tau \sigma {\tau }^{-1})=c_i(\sigma )$.

Let $\mathbb{Q}[X_1,X_2,\dots ]$ be the polynomial ring on the family of indeterminates indexed by $i\ge 1$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials); its elements are finitely supported coefficient families on the monomials of Monomials on an index set as finitely supported exponent families, so every element involves only finitely many of the indeterminates. A character polynomial is an element $P\in \mathbb{Q}[X_1,X_2,\dots ]$. Setting $\mathrm{deg}{X}_i:=i$ gives the monomial $X^a={\prod }_i{X}_i^{a_i}$ the weighted degree ${\sum }_{i}i{a}_i$ (a finite sum); a character polynomial has weighted degree $\le d$ when every monomial occurring in it has weighted degree at most $d$, and the weighted degree of $P\mathrm{\ne }0$ is the largest weighted degree of a monomial of $P$. Products and rational linear combinations of class functions on $S_n$ are again class functions, so every character polynomial defines for each $n\ge 0$ a class function

$P\text{\hspace{0.17em}⁣}:S_n⟶\mathbb{Q},\sigma ⟼P(X_1(\sigma ),X_2(\sigma ),\dots ),$

the defining sum being finite because $P$ has finite support and $X_i(\sigma )=0$ for $i>n$; equivalently, the substitution $X_i\mapsto X_i(\sigma )$ is the ring homomorphism supplied by Universal property of a polynomial ring on an arbitrary family of indeterminates. In this way a character polynomial is realized as one class function on each $S_n$. The realization is not injective on a single $S_n$, since $X_i$ vanishes identically on $S_n$ for $i>n$. It is injective in the limit:

Uniqueness. If $P,Q$ agree as functions on $S_n$ for infinitely many $n$, then $P=Q$.

Indeed, put $R:=P-Q$ and let $A\subseteq \mathbb{N}$ be infinite with $R{\mathrm{\mid }}_{S_n}=0$ for every $n\in A$. Fix $m\ge 0$, $\sigma \in S_m$ and $k\ge 0$, and let ${\sigma }^{(k)}\in S_{m+k}$ be $\sigma$ extended by $k$ fixed points outside its domain; then $X_1({\sigma }^{(k)})=X_1(\sigma )+k$ and $X_i({\sigma }^{(k)})=X_i(\sigma )$ for $i\ge 2$, so $R({\sigma }^{(k)})={\sum }_j(X_1(\sigma )+k{)}^j{R}_j(\sigma )$, where $R={\sum }_j{X}_1^j{R}_j$ collects the monomials of $R$ by their $X_1$-exponent $j$, each $R_j$ involving only $X_2,X_3,\dots$. The right-hand side is given by a polynomial in $k$ over $\mathbb{Q}$ of degree at most the largest $j$ with $R_j\mathrm{\ne }0$, and it vanishes for all $k$ with $m+k\in A$, infinitely many $k$; by the root bound of A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots that polynomial is zero, so $R(\sigma )=0$ (Evaluation and roots of a polynomial in a commutative target ring). As $\sigma \in S_m$ was arbitrary, $R$ vanishes on every $S_m$; equivalently, $R(b_1,\dots ,b_M)=0$ for every $M\ge 0$ and every tuple $(b_1,\dots ,b_M)\in {\mathbb{N}}^M$, since a disjoint union of $b_i$ cycles of length $i$, $1\le i\le M$, is a permutation with $X_i=b_i$ for $i\le M$. If $R\mathrm{\ne }0$, then $R$ involves only $X_1,\dots ,X_M$ for some $M$, and induction on $M$ shows that such an $R$ cannot vanish at every point of ${\mathbb{N}}^M$: for $M=0$ the polynomial is a constant evaluated at the empty tuple; and for $M\ge 1$, fixing $(b_1,\dots ,b_{M-1})\in {\mathbb{N}}^{M-1}$ makes $b_M\mapsto R(b_1,\dots ,b_{M-1},b_M)$ a polynomial in one variable over $\mathbb{Q}$ with infinitely many roots, hence the zero polynomial, so every $X_M$-coefficient of $R$ vanishes at $(b_1,\dots ,b_{M-1})$ and the induction applies to each of those coefficients. Hence $R=0$.

A sequence $({\chi }_n{)}_{n\ge 0}$ of class functions with ${\chi }_n$ a class function on $S_n$ is eventually polynomial when there are $N\ge 0$ and a character polynomial $P$ with

${\chi }_n(\sigma )=P(X_1(\sigma ),X_2(\sigma ),\dots )$

for all $n\ge N$ and all $\sigma \in S_n$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$). By uniqueness in the limit, such a $P$ is determined by $({\chi }_n)$ whenever it exists. No choice principle is used.