Combinatorial criterion for nonzero Schur-expansion coefficients

Determine a simple combinatorial rule characterizing when the Schur-expansion coefficient M_{(μ^(1)|⋯|μ^(r))}^{(λ^(1)|⋯|λ^(r))} of a multi-symmetric Schur function is nonzero.

Background

The paper proves that every multi-symmetric Schur function S_{(λ1|⋯|λr)}(X_1|⋯|X_r) expands positively in the tensor-product Schur basis, with coefficients M_{(μ1|⋯|μr)}{(λ1|⋯|λr)} identified as multiplicities of irreducible representations of a product Levi subgroup inside a Demazure module. Although this establishes nonnegativity and provides a representation-theoretic interpretation, it does not give a direct combinatorial characterization of which coefficients vanish.

References

Theorem \ref{Schur positivity theorem} leads to two interesting questions. First, is there a simple combinatorial rule to determine when $M_{(\mu{(1)}|\cdots | \mu{(r)})}{(\lambda{(1)}|\cdots | \lambda{(r)})} \neq 0?$

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