"Kronecker Product" for quasi-symmetric functions Recall that the Kronecker product
$s_\lambda * s_\mu$ of two Schur functions $s_\lambda$ and $s_\mu$ is the symmetric function
whose expansion (in terms of Schur functions) is given by
\begin{equation}
\sum_{\nu \, \vdash \, n} g_{\lambda \mu}^\nu \, s_\nu
\end{equation}
where $\lambda$, $\mu$, and $\nu$ are partitions of $n$ and
$g_{\lambda \mu}^\nu$ is the Kronecker coefficient, which famously
counts the multiplicity of $V_\nu$ in the tensor product $V_\lambda \otimes V_\mu$ of the irreducible representations of the symmetric group $S_n$.
Switch now to the quasi-symmetric world: Given a composition $\alpha = (\alpha_1, \dots, \alpha_k)$ of $n$ let $L_\alpha$ be the fundamental quasi-symmetric functions defined by
\begin{equation}
L_\alpha = \sum x_{\ell_1} \cdots \,x_{\ell_k}
\end{equation}
where the sum is taken over all sequences
$1 \leq \ell_1 \leq \cdots \leq \ell_k$ such that $\ell_i < \ell_{i+1}$ whenever
$i = \alpha_1 + \cdots + \alpha_j$ for some $1 \leq j \leq k-1$.
The space of symmetric functions within the $\mathrm{QSym}_n :=\Bbb{Q}$-span of $\{ L_\alpha \, \big| \, \alpha \, \models \, n \}$ coincides with the
$\mathrm{Sym}_n:= \Bbb{Q}$-span of $\{ s_\lambda \, \big| \, \lambda \, \vdash \, n \}$
the latter of which is endowed with the Kronecker $*$-product.
Question:
Can the Kronecker $*$-product on $\mathrm{Sym}_n$ be extended to all
of $\mathrm{QSym}_n$ so that there exist non-negative
integers $\tilde{g}_{\alpha,\beta}^{\, \gamma}$ for each
triple $\alpha$, $\beta$, $\gamma$ of compositions of $n$
satisfying
\begin{equation}
L_\alpha * L_\beta = 
\sum_{\gamma \, \models \, n} \tilde{g}_{\alpha,\beta}^{\, \gamma} 
\, L_\gamma \quad \text{?}
\end{equation}
p.s. Covertly, I am asking whether or not there is some kind of tensor product structure (as in a symmetric tensor category) on the projective indecomposable representations of the 0-Hecke algebra $H_n(0)$. Any thoughts on that would also be appreciated.
thanks, ines.
 A: In the world of symmetric functions, Kronecker coefficients give the structure constants for both the inner multiplication and the inner comultiplication. While the natural introduction of the inner multiplication uses the representations/characters of the symmetric group, the inner comultiplication has a very straightforward description: the structure constants are obtained via the expansion
$$s_{\lambda}(x_1y_1,...,x_iy_j,...)=\sum_{\nu ,\mu\, \vdash \, n} g_{\nu \mu}^\lambda \, s_\nu (x)s_{\mu}(y).$$
Similarly we can ask whether $L_{\alpha}(xy)$ has such an expansion, and the answer is positive! There exist nonnegative integers $\tilde{g}_{\beta, \gamma}^{\, \alpha}$ such that
$$L_{\alpha}(xy)=\sum_{\beta, \gamma \, \models \, n}\tilde{g}_{\beta, \gamma}^{\, \alpha}L_{\beta}(x)L_{\gamma}(y).$$
This gives a natural inner comultiplication for quasisymmetric functions. Moreover, by dualizing, these allow us to define a genuine inner multiplication for the dual $\widehat{\mathrm{QSym}_n}$, which is isomorphic to the descent algebra of Solomon.
Originally this was shown in Gessel's paper, where you will find the details and combinatorial interpretations

I. Gessel, "Multipartite P-partitions and inner products of skew
Schur functions", Contemporary Mathematics, 34:289–317, 1984

You can also read about it in

C. Malvenuto, C. Reutenauer
"Duality between quasi-symmetric functions and the Solomon descent algebra"
J. Algebra, 177 (1995), pp. 967-982


Now, as far as a natural inner multiplication for $\mathrm{QSym}$, this is an open problem, at least according to Hazewinkel in

M. Hazewinkel, "Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions", Acta Appl. Math. 75 (2003), 1-3, 55–83

The same paper mentions that there is no known structure on $H_n(0)$ that induces the inner comultiplication from above, so the answer to the last question in the postscript is also missing at the moment.
A: One silly (or super wishful thinking approach)
is to use the formula
$$
g_{\lambda \mu \nu} = \frac{1}{n!} \sum_{\sigma \in S_n} 
\chi^{\lambda}(\sigma)
\chi^{\mu}(\sigma)
\chi^{\nu}(\sigma).
$$
Perhaps some version of
$$
g_{\alpha, \beta,\gamma} = \frac{1}{n!} \sum_{\tau \in S_n} 
\chi^{\alpha}(\tau)
\chi^{\beta}(\tau)
\chi^{\gamma}(\tau)
$$
where now  $\chi^{\alpha}(\tau)$ are the coefficients which show up
when the quasisymmetric power sums are expanded in terms of
the Gessel quasisymmetric functions.
Here, I suppose that in the sum, one does not only consider
the cycle type, but decide to order the cycles by smallest element,
and then let $\alpha$ be the integer composition given by the cycle lengths in that order.
If the coefficients are non-negative by some miracle, then, well, you have something cool to start with.
Ballantine, Cristina; Daugherty, Zajj; Hicks, Angela; Mason, Sarah; Niese, Elizabeth, On quasisymmetric power sums, J. Comb. Theory, Ser. A 175, Article ID 105273, 36 p. (2020). ZBL1442.05241.
