= Induction ring of symmetric-group characters
{title2=$\bigoplus_{n\ge0}R(S_n)$}
The graded abelian group $\bigoplus_{n\ge0}R(S_n)$ of ordinary virtual <characters> has product $\chi\star\psi=\operatorname{Ind}_{S_m\times S_n}^{S_{m+n}}(\chi\boxtimes\psi)$ for degrees $m,n$. Its unit is the trivial degree-zero character. Associativity follows by transitivity of <induced representations> through the three-factor Young subgroup; commutativity follows by conjugating the block positions. The <Frobenius characteristic map> identifies this ring with the ring of <symmetric functions>, sending irreducible Specht characters to <Schur functions>. This product is different from pointwise multiplication of characters of the same symmetric group.
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