The outer induction product of symmetric-group characters

For $m,n\ge 0$, $S_m$ is the symmetric group of $\{1,\dots ,m\}$, with $S_0=\{1\}$ (Young subgroups, tabloids, and permutation modules). Identify $S_m\times S_n$ (The external direct product $G\times H$ with componentwise multiplication) with the subgroup of $S_{m+n}$ preserving each of the two blocks: the first factor acts on $\{1,\dots ,m\}$ and the second on $\{m+1,\dots ,m+n\}$, so that $(\sigma ,\tau )$ acts as $\sigma$ on the first block and as $\tau$ on the second after shifting its labels by $m$. Either block may be empty; its symmetric group is trivial. These actions give an injective homomorphism, and every permutation preserving the blocks has a unique pair of restrictions, so its image is exactly the stated subgroup. For honest characters $\chi$ of $S_m$ and $\psi$ of $S_n$, let $\chi ⊠\psi$ be the character of $S_m\times S_n$ on the tensor product $V{\otimes }_{\mathbb{C}}W$ of representations affording $\chi$ and $\psi$, with $(\sigma ,\tau )\cdot (v\otimes w):=\sigma v\otimes \tau w$, so that

$(\chi ⊠\psi )(\sigma ,\tau )=\chi (\sigma )\psi (\tau )$

(The tensor product of two complex representations, Characters add on direct sums, multiply on tensor products, and conjugate on duals).

Every virtual character $f\in R(S_m)$ is an integral combination $f={\sum }_i{a}_i{\chi }_i$ of the irreducible characters ${\chi }_i$ of $S_m$ (Virtual characters and the character ring $R(G)$ of a finite group), and the coefficients $a_i$ are unique because the irreducible characters of a finite group are orthonormal, hence $\mathbb{Z}$-linearly independent, in $\mathrm{c}\mathrm{f}(S_m)$ (The irreducible complex characters form an orthonormal basis of $\mathrm{c}\mathrm{f}(G)$). So for $f={\sum }_i{a}_i{\chi }_i$ and $g={\sum }_j{b}_j{\psi }_j$ we may set

$f⊠g:={\sum }_{i,j}{a}_i{b}_j({\chi }_i⊠{\psi }_j)\in R(S_m\times S_n),$

an integral combination of honest characters of $S_m\times S_n$. Its value at $(\sigma ,\tau )$ is $f(\sigma )g(\tau )$ by the displayed character formula, and uniqueness of the coefficients makes $f⊠g$ well defined. The assignment is $\mathbb{Z}$-bilinear in $(f,g)$. The outer induction product is

$f\circ g:={\sum }_{i,j}{a}_i{b}_j{\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}({\chi }_i⊠{\psi }_j)\in R(S_{m+n}),$

the induction of honest characters in the sense of The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character. The trivial character of $S_0$ is the unit of degree zero. This is the outer product, defined across different symmetric groups; the same-rank tensor product $f\cdot g$ on a single $R(S_n)$ is a different operation, and the two are never conflated. No choice principle is used.