The Frobenius characteristic map

Fix $n\ge 0$ and let $\mathrm{c}\mathrm{f}(S_n)$ be the complex vector space of class functions on $S_n$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$). For $f\in \mathrm{c}\mathrm{f}(S_n)$ and a partition $\rho ⊢n$ let $f(\rho )$ denote the common value of $f$ on permutations of cycle type $\rho$, which is well defined because cycle type determines the conjugacy class (The conjugacy classes of $S_n$ are indexed by the tuples $(c_1,\dots ,c_n)$ with $\sum k{c}_k=n$). Let

$z_{\rho }:={\prod }_{i\ge 1}{i}^{m_i(\rho )}{m}_i(\rho )!,$

where $m_i(\rho )$ is the number of parts of $\rho$ equal to $i$; this is the order of the centralizer of an element of cycle type $\rho$ (If $\sigma \in S_n$ has $c_k$ cycles of length $k$, then $\mathrm{\mid }{C}_{S_n}(\sigma )\mathrm{\mid }={\prod }_{k=1}^n{k}^{c_k}{c}_k!$), so $z_{\rho }$ is a positive integer. The Frobenius characteristic of $f$ is

$\mathrm{ch}(f):={\sum }_{\rho ⊢n}f(\rho )\frac{p_{\rho }}{z_{\rho }}\in {\mathrm{\Lambda }}_{\mathbb{C}}^n,{\mathrm{\Lambda }}_{\mathbb{C}}:=\mathbb{C}{\otimes }_{\mathbb{Z}}\mathrm{\Lambda }\mathrm{.}$

The sum is finite and well defined because $\{p_{\rho }:\rho ⊢n\}$ is a $\mathbb{Q}$-basis of ${\mathrm{\Lambda }}_{\mathbb{Q}}^n$ (Power sums form a rational but not integral stable basis); equivalently the family $\{p_{\rho }\mathrm{/}{z}_{\rho }:\rho ⊢n\}$ is a $\mathbb{Q}$-basis of ${\mathrm{\Lambda }}_{\mathbb{Q}}^n$ and is orthogonal for the Hall form, with $⟨p_{\rho }\mathrm{/}{z}_{\rho },p_{\sigma }\mathrm{/}{z}_{\sigma }{⟩}_H={\delta }_{\rho \sigma }\mathrm{/}{z}_{\rho }$ (Power sums are orthogonal for the Hall form). The codomain is ${\mathrm{\Lambda }}_{\mathbb{C}}$, not ${\mathrm{\Lambda }}_{\mathbb{Q}}$: for an arbitrary complex class function the coefficients $f(\rho )\mathrm{/}{z}_{\rho }$ are complex. If all values of $f$ lie in $\mathbb{Q}$, then $\mathrm{ch}(f)\in {\mathrm{\Lambda }}_{\mathbb{Q}}^n$; the dictionary theorems proved later on this page show that every virtual character of $S_n$ is rational-valued, and in fact integral-valued, so that its characteristic lies in the integral lattice ${\mathrm{\Lambda }}^n$.

Writing ${\mathrm{c}\mathrm{f}}_S:={⨁}_{n\ge 0}\mathrm{c}\mathrm{f}(S_n)$, the map $\mathrm{ch}:{\mathrm{c}\mathrm{f}}_S\to {\mathrm{\Lambda }}_{\mathbb{C}}$ is defined degreewise by the displayed formula on each $\mathrm{c}\mathrm{f}(S_n)$. It is $\mathbb{C}$-linear on each summand, because evaluation $f\mapsto f(\rho )$ and scalar multiplication are linear. Its restriction to the character ring $R_S={⨁}_{n\ge 0}R(S_n)$ is the Frobenius characteristic dictionary studied in the remaining items of this page. No choice principle is used.