The graded ordinary representation ring of the symmetric groups

For $n\ge 0$ let $R(S_n)$ be the character ring of the finite group $S_n$, the $\mathbb{Z}$-span of its irreducible complex characters (Virtual characters and the character ring $R(G)$ of a finite group), so that $R(S_0)=\mathbb{Z}\cdot \mathbf{1}$ for the trivial group $S_0=\{1\}$ (The finite symmetric group $S_n$, one-line notation, and cycle notation). The graded ordinary representation ring of the symmetric groups is the direct sum

$R_S:={⨁}_{n\ge 0}R(S_n),$

the abelian group of finitely supported tuples $(f_n{)}_{n\ge 0}$ with $f_n\in R(S_n)$ and componentwise addition; an element $f\in R(S_n)$ is homogeneous of degree $n$, and the degree-$n$ component of an element $f={\sum }_n{f}_n$ of $R_S$ is $f_n$. This item defines only the graded abelian group and its degree decomposition: the multiplication used on $R_S$ is the outer induction product of The outer induction product of symmetric-group characters, not the tensor-product multiplication inside a single $R(S_n)$, and no ring axioms for the outer product are assumed here. We write

$R_{S,\mathbb{Q}}:=\mathbb{Q}{\otimes }_{\mathbb{Z}}{R}_S,R_{S,\mathbb{C}}:=\mathbb{C}{\otimes }_{\mathbb{Z}}{R}_S$

for the scalar extensions. No choice principle is used.