TITLE: "Kronecker Product" for quasi-symmetric functions
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.