The integrality conjecture for Jack superpolynomial coefficients

At least 21 years old · documented by 2004

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.

Equivalent formulations 1Other wordings

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).

References

Primary source

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

Posted 1 month ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.