Positivities in Hall–Littlewood expansions and related plethystic operators
DOI:
https://doi.org/10.33044/revuma.2899Abstract
The Hall–Littlewood polynomials $\mathbf{H}_\lambda = Q'_\lambda[X;q]$ are an important symmetric function basis that appears in many contexts. In this work, we give an accessible combinatorial formula for expanding the related symmetric functions $\mathbf{H}_\alpha$ for any composition $\alpha$, in terms of the complete homogeneous basis. We do this by analyzing Jing's operators, which extend to give nice expansions for the related symmetric functions $\mathbf{C}_\alpha$ and $\mathbf{B}_\alpha$ which appear in the formulation of the Compositional Shuffle Theorem. We end with some consequences related to eigenoperators of the modified Macdonald basis.
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