Kronecker coefficients and the internal tensor product

Fix an integer $n\ge 0$ and work over $\mathbb{C}$. Let $\lambda ,\mu ⊢n$ be partitions of $n$ (Partitions, English diagrams, and conjugation), and let $S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }$ be the tensor product of their complex Specht modules, a finite-dimensional complex $S_n$-module (The tensor product of two complex representations, Column antisymmetrizers, polytabloids, and Specht modules). The Kronecker coefficient $g(\lambda ,\mu ,\nu )$ of a third partition $\nu ⊢n$ is the multiplicity of $S^{\nu }$ in this tensor product:

$g(\lambda ,\mu ,\nu ):=\frac{{\dim }_{\mathbb{C}}(S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }{)}_{(S^{\nu })}}{{\dim }_{\mathbb{C}}{S}^{\nu }},$

where $(S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }{)}_{(S^{\nu })}$ is the isotypic component of type $S^{\nu }$ (The isotypic component of a completely reducible representation).

Well-definedness. The module $S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }$ is finite-dimensional, and $\mathrm{char}\mathbb{C}=0$ does not divide the order of the finite group $S_n$, so complex Maschke's theorem gives complete reducibility (Maschke's theorem for finite groups over fields whose characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$, If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, every finite-dimensional representation of $G$ is completely reducible, A completely reducible representation as a finite direct sum of irreducible subrepresentations). The complex Specht modules $S^{\nu }$, $\nu ⊢n$, are a complete list of pairwise inequivalent irreducible $S_n$-modules (Complex Specht modules are irreducible, Specht modules classify the complex irreducibles of $S_n$, Distinct complex Specht modules are inequivalent). Consequently a decomposition into irreducibles has the form $S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }\cong {⨁}_{\nu ⊢n}{m}_{\nu }{S}^{\nu }$ with nonnegative integer multiplicities $m_{\nu }$, and the isotypic component of type $S^{\nu }$ is the direct sum of exactly the $m_{\nu }$ copies of $S^{\nu }$ appearing in any such decomposition, by The isotypic decomposition of a completely reducible representation is unique (Proof step 2.1 identifies each grouped block with the defining isotypic sum). In particular ${\dim }_{\mathbb{C}}(S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }{)}_{(S^{\nu })}=m_{\nu }{\dim }_{\mathbb{C}}{S}^{\nu }$ and ${\dim }_{\mathbb{C}}{S}^{\nu }>0$ because $S^{\nu }$ is irreducible. Hence $g(\lambda ,\mu ,\nu )=m_{\nu }$ is a nonnegative integer, independent of the chosen decomposition, and $g(\lambda ,\mu ,\nu )$ is by definition the multiplicity of $S^{\nu }$ in $S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }$.

The internal product. For partitions $\lambda ,\mu ⊢n$ call $(\lambda ,\mu )⟼S^{\lambda }{\otimes }_{\mathbb{C}}{S}^{\mu }$ the internal (Kronecker) product of the $S_n$-representations $S^{\lambda }$ and $S^{\mu }$. Unlike the outer induction product ${\mathrm{Ind}}_{S_a\times S_b}^{S_{a+b}}(S^{\alpha }⊠S^{\beta })$ of The outer Littlewood–Richardson rule, which takes a pair of modules of degrees $a$ and $b$ to a module of degree $a+b$, the internal product keeps the rank: both inputs and the output are $S_n$-modules of the same degree $n$. The two operations are different constructions on different data; the outer product is mentioned here only for contrast and is not used in this definition.

No form of the Axiom of Choice is used: the tensor product, Maschke's theorem, the Specht classification and the isotypic decomposition are finite-dimensional complex representation theory of a finite group.