The Schur superpolynomial identification conjecture

A superpartition is denoted by [?][?]; let sΛc(x,θ)s^c_\Lambda(x,\theta) be the combinatorially defined Schur superpolynomial and let sΛ(x,θ)s_\Lambda(x,\theta) be the specialization at q=t=0q=t=0 of the Macdonald superpolynomial. Their Kostka coefficients are denoted by KˉΛΩc\bar{K}^c_{\Lambda\Omega} and KˉΛΩ\bar{K}_{\Lambda\Omega}, respectively. Schur superpolynomial identification conjecture.

sΛc(x,θ)=sΛ(x,θ),s^c_\Lambda(x,\theta)=s_\Lambda(x,\theta),

equivalently,

KˉΛΩc=KˉΛΩ.\bar{K}^c_{\Lambda\Omega}=\bar{K}_{\Lambda\Omega}.

This conjecture identifies the combinatorial construction with the Macdonald-superpolynomial specialization and supplies the intended link between the two descriptions; the source gives no resolution status.

Sources & referencesView supporting material

Primary source

Olivier Blondeau-Fournier and Pierre Mathieu, “Schur Superpolynomials: Combinatorial Definition and Pieri Rule”, arXiv:1408.2807 (2015).

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