MathDB

Image conjecture for catabolizable tableaux

At least 27 years old · documented by

Let RR be a sequence of rectangles. Let bR:RC(λt;Rt)→CST(λ,γ(R))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 Im⁡bR\operatorname{Im}b_R denote its image. Let CT(λ;R)CT(\lambda;R) be the set of RR-catabolizable tableaux. Image conjecture. If RR is dominant, then

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

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

References

Primary source

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

rigged-configurationstableaux
Posted

Progress summary

Refreshed
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 Im⁡bR=CT(λ;R)\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 Kλ;R(q)K_{\lambda;R}(q) is obtained as the charge-generating function of RR-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

Solutions 0

No solutions have been posted yet.