Image conjecture for catabolizable tableaux

Let $R$ be a sequence of rectangles. Let $b_R:RC(\lambda^t;R^t)\to CST(\lambda,\gamma(R))$ be the restriction of the inverse rigged-configuration bijection to the indicated subset, and let $\operatorname{Im}b_R$ denote its image. Let $CT(\lambda;R)$ be the set of $R$ -catabolizable tableaux. Image conjecture. If $R$ is dominant, then

\operatorname{Im}b_R=CT(\lambda;R),

and $b_R\circ\Phi_R$ is a bijection $LRT(\lambda;R)\to CT(\lambda;R)$ sending $\operatorname{charge}_R$ to $\operatorname{charge}$ . This would connect rigged configurations, Littlewood–Richardson tableaux, and catabolizable tableaux through a common statistic-preserving map.

Sources & referencesView supporting material

Primary source

Anatol N. Kirillov and Mark Shimozono, “A generalization of the Kostka-Foulkes polynomials”, arXiv:math/9803062 (1998).

rigged-configurations tableaux

1 week ago

Progress summary

Refreshed 6 hours ago

Open

The original work proves only special cases of the conjecture, and no later public proof or counterexample was found.

Shimozono and Weyman’s 1998 work connects generalized Kostka polynomials with catabolizable tableaux and charge. The full identification $\operatorname{Im} b_R=CT(\lambda;R)$ , together with the asserted statistic-preserving bijection, is not established there.

Known results

Shimozono and Weyman, 1998: a general proof strategy leaves open the required sign-reversing involution preserving catabolizability; the conjecture is proved in special cases, including the Kostka–Foulkes case.
Shimozono, 1998: for rectangle sequences linearly ordered by containment, the related conjecture is proved and $K_{\lambda;R}(q)$ is obtained as the charge-generating function of $R$ -catabolizable tableaux.
The retrieved rigged-configuration literature records related polynomial identities as conjectural, not as a proof of the stated image equality.

August 2026 scan

No retrieved source reports a later proof, counterexample, verification, withdrawal, or AI/model claim concerning this exact image conjecture.

Current status (as of August 2026): Special cases and related ordered-rectangle results are known, but the full image equality and the resulting bijection remain open.

Sources

No solutions have been posted yet.