---
title: Star Product of Symmetric Functions
url: https://www.emergentmind.com/topics/star-product-of-symmetric-functions
type: topic
---

# Star Product of Symmetric Functions

The star product of symmetric functions is a bilinear multiplication on the ring of symmetric functions that, in the symmetric-group character setting, encodes pointwise multiplication of class functions and realizes Kronecker-type structure constants inside symmetric-function bases. In the irreducible-character basis $\{\tilde s_\lambda\}$ introduced by Orellana and Zabrocki, the product is characterized by evaluation at roots of unity and has structure coefficients given by the stable Kronecker coefficients $\overline g_{\mu,\nu}^{\,\lambda}$ [1709.08098]. In later literature, the same term also appears in distinct contexts, notably for deformation products on multisymmetric functions and for Kronecker/Hadamard products in alternative bases arising from species; these usages are related by terminology rather than by a single uniform construction [2509.07912], [2604.10336].

## 1. Definition in the irreducible-character basis

Let $\Lambda=\mathbb Q[h_1,h_2,\dots]=\mathbb Q[p_1,p_2,\dots]$ be the ordinary Hopf algebra of symmetric functions. For each partition $\lambda$, the basis element $\tilde s_\lambda\in\Lambda$ is defined by the evaluation property
\[
\tilde s_\lambda[\Xi_\mu]=\chi^{(n-|\lambda|,\lambda)}(\mu)
\]
for all $n\gg 0$ and all cycle-type partitions $\mu\vdash n$, where $\chi^{(n-|\lambda|,\lambda)}$ is the irreducible character of $S_n$ indexed by the stabilized partition $(n-|\lambda|,\lambda)$, and $\Xi_\mu$ denotes the multiset of roots of unity whose cycle-structure is $\mu$ [1709.08098]. Equivalently,
\[
\tilde s_\lambda=\sum_{\nu\vdash |\lambda|}\chi^{(n-|\lambda|,\lambda)}(\nu)\,\frac{p_\nu}{z_\nu}\qquad(n\gg 0).
\]

The star product $\star$ is obtained by transporting the internal (Kronecker) product of class functions on $S_n$ back to $\Lambda$ through the map $f\mapsto f[\Xi_\mu]$. On class functions one has
\[
\chi\cdot\psi(\sigma)=\chi(\sigma)\psi(\sigma),
\]
and in the $\tilde s$-basis this yields
\[
\tilde s_\mu\star \tilde s_\nu=\sum_\lambda \overline g_{\mu,\nu}^{\,\lambda}\,\tilde s_\lambda,
\]
where $\overline g_{\mu,\nu}^{\,\lambda}=\lim_{n\to\infty}g_{(n-|\mu|,\mu),(n-|\nu|,\nu)}^{(n-|\lambda|,\lambda)}$ are the stable Kronecker coefficients [1709.08098].

A parallel formulation appears in the foundational paper on symmetric-group characters as symmetric functions. There, the star product is the unique bilinear operation satisfying
\[
p_\alpha\star p_\beta=\delta_{\alpha,\beta}\,z_\alpha\,p_\alpha,
\]
or equivalently
\[
\langle f\star g,p_\mu\rangle=\langle f,p_\mu\rangle\,\langle g,p_\mu\rangle,
\]
and the same operation expands on the $\tilde s$-basis by stable Kronecker coefficients [1605.06672].

## 2. Algebraic structure and representation-theoretic meaning

The star product is commutative and associative because pointwise multiplication of class functions on a finite group is commutative and associative [1709.08098]. In the irreducible-character basis it is therefore a second multiplication on $\Lambda$, distinct from the ordinary product of symmetric functions.

A second scalar product $\langle-,-\rangle_{@}$ is defined by choosing $n\gg 0$ and setting
\[
\langle f,g\rangle_{@}=\frac1{n!}\sum_{\sigma\in S_n}f[\Xi_{\mathrm{cyc}(\sigma)}]\,g[\Xi_{\mathrm{cyc}(\sigma)}].
\]
For sufficiently large $n$, this is independent of $n$, and the basis $\{\tilde s_\lambda\}_\lambda$ is orthonormal:
\[
\langle \tilde s_\lambda,\tilde s_\mu\rangle_{@}=\delta_{\lambda\mu}.
\]
With respect to this inner product, the star product is adjoint to the usual coproduct on $\Lambda$; equivalently,
\[
\langle f\star g,h\rangle_{@}=\langle f,g\star h\rangle_{@}\qquad\forall f,g,h\in\Lambda
\]
[1709.08098].

The basis $\{\tilde s_\lambda\}$ admits three equivalent characterizations. First, it is the unique inhomogeneous basis of $\mathrm{Sym}$ with $\deg \tilde s_\lambda=|\lambda|$ satisfying the evaluation property at roots of unity. Second, for $n\gg 0$, if the $GL_n$-irreducible $W^\lambda$ with character $s_\lambda$ is restricted to $S_n$, then
\[
W^\lambda\downarrow_{S_n}\cong \bigoplus_\mu \bigl(S^{(n-|\mu|,\mu)}\bigr)^{\oplus r_{\lambda\mu}},
\qquad
s_\lambda=\sum_\mu r_{\lambda\mu}\,\tilde s_\mu.
\]
Third, the basis is determined by the initial condition
\[
s_{1^r}=\tilde s_{1^r}+\tilde s_{1^{r-1}}\qquad(r\ge 1)
\]
together with the requirement that the star product have stable Kronecker coefficients as structure constants [1605.06672].

The representation-theoretic content is explicit: $\overline g_{\mu,\nu}^{\,\lambda}$ is the stabilized multiplicity of $S^{(n-|\lambda|,\lambda)}$ in
\[
S^{(n-|\mu|,\mu)}\otimes S^{(n-|\nu|,\nu)}
\]
for $n\gg 0$ [1605.06672]. This identifies the star product in the $\tilde s$-basis with the stable regime of the ordinary Kronecker product of Schur functions having a large first row.

## 3. Tableaux models and Pieri-type rules

A central contribution of Orellana and Zabrocki is a collection of tableaux models for special families of star-product coefficients. The paper does not give a positive combinatorial rule for all $\overline g_{\mu,\nu}^{\,\lambda}$; that problem is stated as still open. Instead, it gives three tableaux-based rules for cases in which one factor is either a complete-homogeneous product or a product of $\tilde s$-basis elements of one-row shape [1709.08098].

The common combinatorial object is a multiset-valued tableau $T$ of shape $(r,\gamma)/(\gamma_1)$, defined as a column-strict filling in French notation whose entries are nonempty multisets from $\{1,2,\dots,\bar1,\bar2,\dots\}$, each containing at most one barred element. No cell in the top row may contain only barred elements. The reading word $\mathrm{read}(T|S)$ selects the barred labels from cells whose unbarred-multiset is exactly $S$, reading each row right to left from bottom to top. If $S_1<\cdots<S_d$ are listed in increasing reverse-lex order, then $T$ is called lattice when
\[
\mathrm{read}(T|\varnothing)\,\mathrm{read}(T|S_1)\,\mathrm{read}(T|S_2)\cdots \mathrm{read}(T|S_d)
\]
is a lattice word, meaning that every prefix has the number of $i$'s at least the number of $(i+1)$'s. The set of such tableaux with unbarred-multiset-content $\alpha$ is denoted $\mathrm{MCT}(\lambda,\gamma;\alpha)$ [1709.08098].

Three counting theorems organize the relevant coefficients. For any partition $\lambda$ and composition $\alpha$, the coefficient of $\tilde s_\gamma$ in
\[
h_{\alpha_1}h_{\alpha_2}\cdots h_{\alpha_\ell}\,\tilde s_\lambda
\]
is exactly the number of tableaux in $\mathrm{MCT}(\lambda,\gamma;\alpha)$. If the unbarred multisets are required to be sets, then the same tableaux model counts the coefficient of $\tilde s_\gamma$ in the same expression viewed as a product of induced Young-subgroup characters. Finally, for genuine star-products of one-row shapes,
\[
\tilde s_{(\alpha_1)}\star\tilde s_{(\alpha_2)}\star\cdots\star\tilde s_{(\alpha_\ell)}\star\tilde s_\lambda,
\]
the same set-valued tableaux are used, with the additional restriction that no singleton unbarred-set may appear in the first row; the number of such lattice tableaux is the coefficient of $\tilde s_\gamma$ [1709.08098].

These results imply three Pieri-type rules. In particular,
\[
h_k\star \tilde s_\lambda
\]
expands by multiset-valued lattice tableaux with one $k$-fold unbarred entry,
\[
\tilde s_{(k)}\star \tilde s_\lambda
\]
expands by set-valued lattice tableaux whose unbarred-content is one $k$-set, and, more generally,
\[
\tilde s_{(\alpha_1)}\star\cdots\star\tilde s_{(\alpha_\ell)}\star\tilde s_\lambda
\]
expands by set-valued lattice tableaux whose unbarred-labels form sets of sizes $\alpha_1,\dots,\alpha_\ell$ with no singleton in the first row [1709.08098].

## 4. Worked examples and applications

Two small examples illustrate the stable Kronecker interpretation. For $\mu=(2)$ and $\nu=(1,1)$, the stable decomposition of the Kronecker product of $S_n$-characters $\chi^{(n-2,2)}\otimes\chi^{(n-2,1,1)}$ is
\[
\chi^{(n-2,2)}\oplus\chi^{(n-2,1,1)},
\]
and therefore
\[
\tilde s_{(2)}\star\tilde s_{(1,1)}=\tilde s_{(2)}+\tilde s_{(1,1)}.
\]
A single-box rule is
\[
\tilde s_{(1)}\star\tilde s_\lambda
=\sum_{\gamma:\,\gamma/\lambda\ \text{a horizontal strip of size }1}\tilde s_\gamma,
\]
because multiplication by the trivial Young-subgroup of type $(n-1,1)$ amounts to branching by removing or adding a corner [1709.08098].

The change-of-basis coefficients between the ordinary Schur basis $s_\lambda$ and the basis $\tilde s_\mu$ are the decomposition multiplicities appearing in the restriction of an irreducible polynomial $GL_n$-module to $S_n$ [1709.08098]. This makes the basis particularly useful in translating between polynomial representation theory of $GL_n$ and ordinary representation theory of the symmetric group.

Further applications include partition algebras and quantum information. The coefficient of $\tilde s_{(r)}$ in $(h_1)^{\otimes k}\star \tilde s_{(r)}$ is the dimension of the irreducible module of the partition algebra $P_k(r)$, and this coefficient is naturally counted by the same set-valued tableaux; in the classical Bratteli-diagram approach they become the usual vacillating tableaux. In another application, $k$-fold Kronecker products of the form
\[
s_{(d,d)}^{\otimes a}\otimes s_{(d+1,d-1)}^{\otimes(k-a)}
\]
correspond to measuring entanglement of $k$ qubits, and rewriting each Schur factor as a small $h\star\tilde s$-product yields a direct tableaux-counting rule for the multiplicities [1709.08098].

The basis also supports expansion formulas tied to plethysm and Frobenius maps. One has
\[
f=\sum_\lambda \langle f,\tilde s_\lambda\rangle_{@}\,\tilde s_\lambda
\]
for any $f\in\Lambda$ [1709.08098].

## 5. Quantum star products on multisymmetric functions

In a different line of work, the star product of symmetric functions refers to a deformation product on multisymmetric functions. The setting is the algebra of polynomial functions on $\mathbb R^2$ with standard Poisson bracket $\{x,y\}=1$, followed by passage to $S_n$-invariants, yielding the ring of quantum symmetric functions
\[
(K[x_1,\dots,x_n,y_1,\dots,y_n][[\hbar]],\star)^{S_n}.
\]
Here the product is the Moyal–Weyl star product rather than the Kronecker product of symmetric-group characters [2509.07912].

For $f,g\in K[x,y][[\hbar]]$, the star product is
\[
f\star g=\sum_{m=0}^\infty \frac1{m!}(\pi^m\cdot f\otimes g)\,\hbar^m,
\]
and for monomials,
\[
(x^c y^d)\star(x^f y^g)
=\sum_{k=0}^{\min(d,f)}
\mathrm{Binomial}(d,k)\,(f)_k\,x^{c+f-k}y^{d+g-k}\hbar^k.
\]
Passing to invariants and using elementary multisymmetric functions $e_\alpha(p)$, Pariguan–Sierra’s formula is
\[
e_\alpha(p)\star e_\beta(q)
=\sum_{m=0}^\infty\ \sum_{\Gamma\in Q(\alpha,\beta,n,m)} e_\Gamma(B(p,q))\,\hbar^m,
\]
where $Q(\alpha,\beta,n,m)$ is a set of cubical matrices determined by marginal and weight conditions [2509.07912].

The parameter $\hbar$ controls the deformation: modulo $\hbar$, the star product reduces to the classical commutative product, and $Q(\alpha,\beta,n,0)=L(\alpha,\beta,n)$, the classical contingency-matrix set. Higher powers of $\hbar$ record the number of Poisson-bracket contractions [2509.07912].

The combinatorics is governed by RSK. Classical contingency matrices correspond to pairs of semistandard tableaux of common shape. In the quantum case, the relevant objects are 3-words, and the key bijection is
\[
\{\Gamma\in Q(\alpha,\beta,n,m)\}\cong \{\text{3-words }\Omega\text{ of total weight }m\}.
\]
This reduces the structure constants of the quantum star product to a level-decorated RSK analysis. The same data also identifies $Q(\alpha,\beta,n,m)$ with integer points in a 3-dimensional transportation polytope [2509.07912]. A plausible implication is that, in this setting, the star product is best viewed as a combinatorial deformation of the classical product of multisymmetric functions rather than as a stable Kronecker operation.

## 6. Kronecker/Hadamard star products in alternative bases

A further development studies the Kronecker, or Hadamard, product in bases coming from combinatorial species. For each partition $\alpha\vdash n$, two species $C_\alpha$ and $K_\alpha$ produce cycle-index symmetric functions
\[
F_\alpha(z)=Z_{C_\alpha}(z),\qquad G_\alpha(z)=Z_{K_\alpha}(z),
\]
and the families $\{F_\alpha:\alpha\vdash n\}$ and $\{G_\alpha:\alpha\vdash n\}$ form $\mathbb Q$-bases of $\Lambda_n$ because their transition matrices to the power-sum basis are upper-triangular with nonzero diagonal entries [2604.10336].

In this degreewise setting, if
\[
f(z)=\sum_{\mu\vdash n} f_\mu p_\mu(z),\qquad
g(z)=\sum_{\mu\vdash n} g_\mu p_\mu(z),
\]
then the Kronecker product is
\[
(f\star g)(z)=\sum_{\mu\vdash n} f_\mu g_\mu z_\mu\,p_\mu(z).
\]
Equivalently, for $\tilde p_\mu=p_\mu/z_\mu$ one has $\tilde p_\mu\star \tilde p_\nu=\delta_{\mu,\nu}\tilde p_\mu$. In particular,
\[
h_\mu(z)\star h_\nu(z)=\delta_{\mu,\nu}\,h_\mu(z)
\]
[2604.10336].

The new bases interact nontrivially with this product. One has
\[
F_\alpha(z)\star F_\beta(z)=\sum_{\gamma\vdash n} b_{\alpha,\beta}^{\gamma}\,F_\gamma(z),
\qquad
G_\alpha(z)\star G_\beta(z)=\sum_{\gamma\vdash n} j_{\alpha,\beta}^{\gamma}\,G_\gamma(z),
\]
where the coefficients are described by double-coset counts and are manifestly nonnegative integers. The associated subcategories of species generated by the homogeneous, first-kind cyclic, and second-kind cyclic molecules are closed under $\star$ [2604.10336].

Examples at $n=6$ show the contrast with the homogeneous basis. For instance,
\[
C_6\star C_6
=18\,C_{1^6}+2\,C_{2^3}+2\,C_{3^2}+2\,C_6,
\]
while
\[
h_6\star h_6=h_6,\qquad h_6\star h_{4,2}=0.
\]
This suggests that the positivity of structure constants can depend strongly on the choice of basis, even when the underlying operation is the same Kronecker/Hadamard product [2604.10336].

## 7. Scope, distinctions, and open directions

The expression “star product of symmetric functions” therefore has at least three technically distinct meanings in current usage. In the Orellana–Zabrocki framework, it is the character-theoretic multiplication on $\Lambda$ whose structure coefficients in the $\tilde s$-basis are the stable Kronecker coefficients [1709.08098]. In the multisymmetric-function setting, it is a formal deformation product built from the Moyal–Weyl formula and indexed combinatorially by cubical matrices, 3-words, and transportation polytopes [2509.07912]. In species-theoretic work, it is the Kronecker/Hadamard product studied in bases where nonnegative integer structure constants arise from molecular decompositions and double-coset counts [2604.10336].

A recurrent misconception is that these are merely different presentations of one operation. The available results do not support that identification. The character-theoretic star product is tied to symmetric-group characters, roots-of-unity evaluation, and stable Kronecker coefficients; the quantum product depends on $\hbar$ and Poisson contractions; the species-theoretic product is degreewise Hadamard multiplication on Frobenius characteristics. What they share is a common role as alternative multiplications on algebras of symmetric or multisymmetric functions.

The principal unresolved issue highlighted in the character-theoretic literature is the absence of a positive combinatorial rule for all stable Kronecker coefficients $\overline g_{\mu,\nu}^{\,\lambda}$ [1709.08098]. The partial tableaux rules, the restriction-from-$GL_n$ interpretation, and the species-theoretic positivity phenomena all suggest that basis choice and categorical origin are central to the problem. A plausible implication is that further progress may come from constructions that, like the $\tilde s$-basis or the species bases $F_\alpha$ and $G_\alpha$, align the star product with an intrinsic combinatorial model rather than with the power-sum basis alone.

Source: https://www.emergentmind.com/topics/star-product-of-symmetric-functions