Kronecker Coefficients and Internal Products

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.

For every $n\ge 0$ and $\lambda ⊢n$ the complex Specht module $S^{\lambda }$ carries a nondegenerate symmetric $S_n$-invariant bilinear form $b(x,y)={\sum }_T{x}_T{y}_T$ given by the coefficientwise product in the tabloid basis, so $v\mapsto b(v,-)$ is an isomorphism of $\mathbb{C}{S}_n$-modules $S^{\lambda }\to (S^{\lambda }{)}^{\ast }$; equivalently $S^{\lambda }$ is self-dual. Consequently ${\chi }^{\lambda }(g^{-1})={\chi }^{\lambda }(g)$ for every $g\in S_n$, that is, every irreducible character of $S_n$ is real-valued.

For all partitions $\lambda ,\mu ,\nu ⊢n$, $g(\lambda ,\mu ,\nu )=⟨{\chi }^{\lambda }{\chi }^{\mu },{\chi }^{\nu }⟩=\frac{1}{n!}{\sum }_{\sigma \in S_n}{\chi }^{\lambda }(\sigma ){\chi }^{\mu }(\sigma ){\chi }^{\nu }(\sigma ),$ where ${\chi }^{\lambda }{\chi }^{\mu }$ is the pointwise product of class functions and the sum uses the real-valuedness of the irreducible characters, so that no conjugation appears on the third factor.

For all partitions $\lambda ,\mu ,\nu ⊢n$:

(a) $g(\lambda ,\mu ,\nu )$ is a nonnegative integer;

(b) $g$ is invariant under every permutation of the three arguments, that is $g(\lambda ,\mu ,\nu )=g(\mu ,\lambda ,\nu )=g(\lambda ,\nu ,\mu )=\cdots$ for all six orderings;

(c) conjugating exactly two of the three partitions preserves the coefficient: $g(\lambda ,\mu ,\nu )=g({\lambda }^{\mathrm{\prime }},{\mu }^{\mathrm{\prime }},\nu )=g({\lambda }^{\mathrm{\prime }},\mu ,{\nu }^{\mathrm{\prime }})=g(\lambda ,{\mu }^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }}),$ where ${\lambda }^{\mathrm{\prime }}$ denotes the conjugate partition.

No positive combinatorial rule for the coefficients is asserted.

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.

For partitions $\rho ,\sigma ⊢n$ one has $p_{\rho }\ast p_{\sigma }={\delta }_{\rho ,\sigma }{z}_{\rho }{p}_{\rho },z_{\rho }={\prod }_{i\ge 1}{i}^{m_i(\rho )}{m}_i(\rho )!,$ where $m_i(\rho )$ is the number of parts of $\rho$ equal to $i$. Consequently $s_{\lambda }\ast s_{\mu }={\sum }_{\nu ⊢n}g(\lambda ,\mu ,\nu )s_{\nu }$ for all $\lambda ,\mu ⊢n$. The internal product on ${\mathrm{\Lambda }}^n$ is therefore diagonal in the power-sum basis after extension to ${\mathrm{\Lambda }}_{\mathbb{Q}}^n$, with $z_{\rho }$ the diagonal scalars, and it has the integer structure constants $g(\lambda ,\mu ,\nu )$ in the Schur basis.

Let $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ be a partition with $\mathrm{\mid }\alpha \mathrm{\mid }={\alpha }_1+\cdots +{\alpha }_k$ (Partitions, English diagrams, and conjugation), and let $n\ge 0$ be an integer. Read $\mathrm{\mid }\mathrm{\varnothing }\mathrm{\mid }=0$ and ${\alpha }_1=0$ when $\alpha =\mathrm{\varnothing }$. Whenever

$n\ge \mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1,$

define the first-row padding of $\alpha$ to size $n$ to be $\alpha [n]:=(n-\mathrm{\mid }\alpha \mathrm{\mid },{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)\mathrm{.}$

The threshold is exact.

Above the threshold. Suppose $n\ge \mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$. Then $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge {\alpha }_1\ge 0$, so every entry of the displayed sequence is a nonnegative integer, the entries sum to $(n-\mathrm{\mid }\alpha \mathrm{\mid })+\mathrm{\mid }\alpha \mathrm{\mid }=n$, and the sequence is weakly decreasing because $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge {\alpha }_1$ and $\alpha$ is weakly decreasing (Partitions, English diagrams, and conjugation). If $\alpha$ is nonempty then ${\alpha }_1\ge 1$, hence $n-\mathrm{\mid }\alpha \mathrm{\mid }\ge 1$ and the displayed sequence is a partition of $n$ with positive parts. If $\alpha =\mathrm{\varnothing }$ then the displayed sequence is $(n)$; this is a partition of $n$ for every $n\ge 1$, while for $n=0$ one takes $\mathrm{\varnothing }[0]:=\mathrm{\varnothing }$, the unique partition of $0$. Thus the padding is defined and is a partition of $n$ on the whole displayed range.

Below the threshold. If $\alpha$ is nonempty and $n<\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$, then $n-\mathrm{\mid }\alpha \mathrm{\mid }<{\alpha }_1$; if also $n\ge \mathrm{\mid }\alpha \mathrm{\mid }$ the leading entry is nonnegative but smaller than the second entry ${\alpha }_1$, and if $n<\mathrm{\mid }\alpha \mathrm{\mid }$ it is negative. In both cases the displayed sequence is not a partition, so $\alpha [n]$ is left undefined. The threshold $\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$ is therefore the exact size at which the padded family begins: for nonempty $\alpha$, at $n=\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1$ the padding equals $({\alpha }_1,{\alpha }_1,{\alpha }_2,\dots )$, which is a partition with a repeated first part when $\alpha$ is nonempty.

Neutrality of the notation. This definition records a family of partitions; it asserts no constancy of any quantity along the family and no stability property. The same notation is used by the FI-module and representation-stability developments; no FI-module structure or generation degree is asserted here.

Let $\alpha ,\beta ,\gamma$ be partitions (Partitions, English diagrams, and conjugation) and put $n_0:=\max \{\mathrm{\mid }\alpha \mathrm{\mid }+{\alpha }_1,\mathrm{\mid }\beta \mathrm{\mid }+{\beta }_1,\mathrm{\mid }\gamma \mathrm{\mid }+{\gamma }_1\},$ where the first part of the empty partition is read as $0$. For every integer $n\ge n_0$ the three first-row paddings $\alpha [n]$, $\beta [n]$, $\gamma [n]$ are partitions of $n$, by First-row padding of a partition and the choice of $n_0$ as the maximum of the three thresholds. On that range define the size-indexed padded Kronecker coefficients by $g_n(\alpha ,\beta ,\gamma ):=g(\alpha [n],\beta [n],\gamma [n]),$ the Kronecker coefficient of three partitions of $n$ (Kronecker coefficients and the internal tensor product).

Domain. The threshold $n_0$ is exactly where the common domain of the three padded families starts: for $n\ge n_0$ all three padded sequences are partitions of $n$, while for $n<n_0$ at least one of the three padded sequences is not a partition of $n$ and the expression is left undefined. The padded Kronecker coefficient sequence, indexed from $0$ on $\mathbb{N}$, is $k⟼g_{n_0+k}(\alpha ,\beta ,\gamma )(k\in \mathbb{N})\mathrm{.}$ Every term is defined since $n_0+k\ge n_0$. The notation $g_n$ retains $n$ as the common size of the padded partitions: it is defined only for $n\ge n_0$, and no values at smaller sizes are introduced by reindexing.

No constancy is asserted. This definition introduces notation only. It asserts neither that $g_n(\alpha ,\beta ,\gamma )$ is eventually constant, nor any bound on it, nor any comparison between different values of $n$; Murnaghan stability is a theorem about this sequence proved elsewhere, and this definition is available as its noncircular input.

Let $\lambda ⊢n$, $\tau ⊢r$ and $\theta ⊢n-r$. Then, in ${\mathrm{\Lambda }}^n$, $s_{\lambda }\ast (s_{\tau }{s}_{\theta })={\sum }_{\alpha ⊢r,\beta ⊢n-r}{c}_{\alpha \beta }^{\lambda }(s_{\alpha }\ast s_{\tau })(s_{\beta }\ast s_{\theta }),$ where $c_{\alpha \beta }^{\lambda }$ is the Littlewood–Richardson coefficient. Both sides have degree $n$ and the sum is finite.