The internal (Kronecker) product on symmetric functions

Let $\mathrm{ch}:R_S\to \mathrm{\Lambda }$ be the Frobenius characteristic restricted to the graded ordinary representation ring of the symmetric groups (The Frobenius characteristic map, The graded ordinary representation ring of the symmetric groups). By The Frobenius characteristic is an isometric graded ring isomorphism it is a degree-preserving $\mathbb{Z}$-module isomorphism onto the stable symmetric function ring $\mathrm{\Lambda }={⨁}_{m\ge 0}{\mathrm{\Lambda }}^m$, so its restriction $\mathrm{ch}:R(S_m)\to {\mathrm{\Lambda }}^m$ is an isomorphism for every $m\ge 0$.

For $f\in {\mathrm{\Lambda }}^m$ and $g\in {\mathrm{\Lambda }}^n$ define $f\ast g:=\{\begin{array}{ll}\mathrm{ch}({\mathrm{ch}}^{-1}(f)\cdot {\mathrm{ch}}^{-1}(g)),& m=n,\\ 0,& m\mathrm{\ne }n,\end{array}$ where $\cdot$ is the pointwise (internal tensor) product of class functions on $S_m$; extend $\ast$ bilinearly to all of $\mathrm{\Lambda }$. Call $\ast$ the internal (Kronecker) product of symmetric functions.

Well-definedness. For $m=n$ the class functions ${\mathrm{ch}}^{-1}(f)$ and ${\mathrm{ch}}^{-1}(g)$ lie in $R(S_m)$, the integral span of the irreducible characters of $S_m$ (The graded ordinary representation ring of the symmetric groups). The pointwise product of two honest characters of $S_m$ is again an honest character, namely the character of the tensor product of the underlying representations (Characters add on direct sums, multiply on tensor products, and conjugate on duals), and products of $\mathbb{Z}$-combinations of characters are computed bilinearly; hence the pointwise product of two elements of $R(S_m)$ lies in $R(S_m)$, and applying the isomorphism $\mathrm{ch}$ lands in ${\mathrm{\Lambda }}^m$. The product is therefore well defined and takes ${\mathrm{\Lambda }}^m\times {\mathrm{\Lambda }}^m$ to ${\mathrm{\Lambda }}^m$.

Basic properties. The product is bilinear by construction and inherits commutativity and associativity from the pointwise product of class functions. It preserves degrees, in the sense that ${\mathrm{\Lambda }}^m\ast {\mathrm{\Lambda }}^n\subseteq {\mathrm{\Lambda }}^m$ for $m=n$ and is $0$ for $m\mathrm{\ne }n$. On ${\mathrm{\Lambda }}^n$ the unit is $h_n=s_{(n)}=\mathrm{ch}(\text{trivial character of }{S}_n)$ for $n>0$, and in degree zero it is $h_0=s_{\mathrm{\varnothing }}=1$, because the trivial character is the unit for pointwise multiplication on $R(S_n)$ (The characteristic of a Specht character is a Schur function identifies $\mathrm{ch}({\chi }^{(n)})=s_{(n)}=h_n$). The graded direct sum $\mathrm{\Lambda }$ has no global unit for $\ast$: if $e\in \mathrm{\Lambda }$ had finite degree support and $n$ were a degree with $e_n=0$ while ${\mathrm{\Lambda }}^n\mathrm{\ne }0$, then $(e\ast h_n{)}_n=e_n\ast h_n=0\mathrm{\ne }{h}_n$, so $e\ast h_n\mathrm{\ne }{h}_n$; a global unit would require $e_n$ to be a unit of $({\mathrm{\Lambda }}^n,\ast )$ for every $n$. In particular the ordinary product unit $1\in {\mathrm{\Lambda }}^0$ is not a unit for $\ast$ on the positive degrees.

Schur expansion. For partitions $\lambda ,\mu ⊢n$, the definition and the Specht character dictionary $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ (The characteristic of a Specht character is a Schur function) give $s_{\lambda }\ast s_{\mu }=\mathrm{ch}({\chi }^{\lambda }\cdot {\chi }^{\mu })=\mathrm{ch}({\sum }_{\nu ⊢n}g(\lambda ,\mu ,\nu ){\chi }^{\nu })={\sum }_{\nu ⊢n}g(\lambda ,\mu ,\nu )s_{\nu },$ because ${\chi }^{\lambda }{\chi }^{\mu }$ is the character of $S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }$ and its decomposition has multiplicities $g(\lambda ,\mu ,\nu )$ (Kronecker coefficients and the internal tensor product).

No form of the Axiom of Choice is used: the transport goes through the choice-free Frobenius characteristic isomorphism and finite-dimensional representation theory of finite groups.