Positive combinatorial formula for Schur-expansion coefficients

Derive an explicit positive combinatorial formula for the coefficients M_{(μ^(1)|⋯|μ^(r))}^{(λ^(1)|⋯|λ^(r))} in the Schur expansion of multi-symmetric Schur functions, necessarily generalizing Littlewood–Richardson coefficients.

Background

The coefficients M_{(μ1|⋯|μr)}{(λ1|⋯|λr)} are shown to be nonnegative integers and to equal Levi-subgroup multiplicities in finite-dimensional Demazure modules. The paper emphasizes that a positive combinatorial formula would have to extend the classical Littlewood–Richardson theory, because when one component of the multipartition is empty, the coefficients specialize to Littlewood–Richardson coefficients in the identity S_{(∅|λ)}(X|Y)=s_λ(X+Y).

References

Second, is there an explicit (positive) combinatorial formula for the $M_{(\mu{(1)}|\cdots | \mu{(r)})}{(\lambda{(1)}|\cdots | \lambda{(r)})}$ coefficients? We caution that any answer to this second question will necessarily involve generalizing the Littlewood-Richardson coefficients $c_{\mu,\nu}{\lambda}$ since even for $r=2$ we have that $$\cS_{(\emptyset|\lambda)}(X|Y) = s_{\lambda}(X+Y) = \sum_{\nu,\nu} c_{\mu,\nu}{\lambda} s_{\mu}(X)s_{\nu}(Y).$$

— Multi-Symmetric Schur Functions  (2502.08738 - Weising, 12 Feb 2025) in Remark 5.1 (labeled “LR remark”), Section 5, “Schur Expansion”