The integrality conjecture for Jack superpolynomial coefficients

From papers

Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, and write its supermonomial expansion in the form

J_\Lambda^{(1/\beta)}=c_\Lambda^{\mathrm{min}}(\beta)\sum_{\Omega\leq\Lambda}\tilde c_{\Lambda\Omega}(\beta)\,\tilde m_\Omega.

Here $\Omega\leq\Lambda$ denotes the ordering used for the triangular expansion, and $\tilde m_\Omega$ are supermonomial functions. Integrality conjecture. Each coefficient $\tilde c_{\Lambda\Omega}(\beta)$ is a polynomial in $1/\beta$ with integral coefficients. This conjecture asserts an arithmetic refinement of the triangular expansion despite the lack of a normalization giving positive supermonomial coefficients.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The integrality conjecture for Jack superpolynomial coefficients
Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, let $c_\Lambda^{\mathrm{min}}(\beta)$ be its minimal coefficient, and expand it in the monomial basis as
$J_\Lambda^{(1/\beta)}=c_\Lambda^{\mathrm{min}}(\beta)\sum_{\Omega\leq\Lambda}\tilde c_{\Lambda\Omega}(\beta)\,\tilde m_\Omega.$
The integrality conjecture. For every $\Omega\leq\Lambda$ , $\tilde c_{\Lambda\Omega}(\beta)$ is a polynomial in $1/\beta$ with integral coefficients. The claim concerns the normalized monomial coefficients of Jack superpolynomials, whose coefficients cannot generally be normalized to be all positive.
source: Patrick Desrosiers, Luc Lapointe and Pierre Mathieu, “Orthogonality of Jack polynomials in superspace”, arXiv:math-ph/0509039 (2006).

Sources & referencesView supporting material

Primary source

P. Desrosiers, L. Lapointe and P. Mathieu, “Symmetric functions in superspace”, arXiv:math/0412306 (2004).

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