Charge on LS galleries and paths

Definitions and mathematical statement

The workshop asks for a charge statistic on galleries that restricts, under the natural injection of tableaux into galleries, to the usual tableau charge. Since galleries can also be viewed as Littelmann paths, it asks for a possible path version as well. The root system, gallery type, finiteness conditions and required additional properties of the statistic are not specified. In particular, the original wording does not explicitly require a Kostka--Foulkes generating function.

The supplied aid separates the following type-$A$ crystal formulation. Let $B(\lambda)$ be the highest-weight crystal of a partition $\lambda$ for $\operatorname{GL}_r$, and let $B(\lambda)_\mu$ denote its fiber at a dominant weight $\mu$. The semistandard tableaux, Lakshmibai--Seshadri galleries and LS paths in the relevant models realize the same crystal. A strict crystal isomorphism preserves weights and commutes with every simple raising and lowering operator $e_i,f_i$.

The aid attributes to Patimo a charge formula, for dominant endpoint, \[ \operatorname{ch}(b)=\sum_{\alpha\in\Phi^+}\varepsilon_\alpha(b),\qquad b\in B(\lambda)_\mu, \] where $\varepsilon_\alpha$ are the modified-root statistics in that construction, and $\Phi^+$ is the positive-root set of type $A_{r-1}$. These statistics must not be replaced without justification by an unspecified ordinary root-string convention. Their definitions and normalization are not supplied here. The reported strict-crystal invariance means \[ \operatorname{ch}(\Psi(b))=\operatorname{ch}(b) \] for the relevant strict isomorphisms $\Psi$. Transport therefore defines intrinsic statistics on the corresponding LS galleries and LS paths, agrees with Lascoux--Schützenberger tableau charge, and gives the stated identity \[ K_{\lambda,\mu}(q)=\sum_{b\in B(\lambda)_\mu}q^{\operatorname{ch}(b)}. \] This assertion is restricted to the stated type-$A$, dominant-endpoint setting and its appropriate crystal models.

A different interpretation would use a strictly larger finite ambient gallery fiber $G_{\lambda,\mu}$ containing the tableau set $T_{\lambda,\mu}$ by an injection $\iota$. The aid's obstruction is to imposing both \[ c(\iota(T))=\operatorname{charge}(T),\qquad \sum_{g\in G_{\lambda,\mu}}q^{c(g)}=K_{\lambda,\mu}(q) \] with one monomial of coefficient $1$ for every gallery. Indeed, specialization at $q=1$ would require $|G_{\lambda,\mu}|=|T_{\lambda,\mu}|$. Equivalently, the tableau image already contributes the full polynomial, and extra nonnegative monomial contributions cannot disappear. This is not an obstruction to merely extending the tableau statistic as a set function, or to different weighted, signed or restricted enumerations. The aid reports only partial progress in general root systems, including advances in type $C$ and $C_2$, without full formulations or references.

Short English statement

Extend tableau charge to galleries or Littelmann paths. The supplied aid reports an intrinsic type-$A$ statistic on the corresponding LS crystal models and explains why a strictly larger finite gallery set cannot enumerate the same Kostka--Foulkes polynomial by one positive monomial per gallery.

Sources

• Local UnsolvedMath registry, permanent ID $20002910$, source code catalogue entry; canonical aim-special-functions-notes.json, notes[14].
• AIM workshop Generalized Kostka polynomials, Problem (15): https://aimath.org/WWN/kostka/prob2.pdf. The supplied verification reports comparison with the workshop page, contemporaneous report and neighboring records, and says a participant proposed the Littelmann-path direction after a galleries presentation. No OCR error, original remarks or literature field are reported.
• The research aid names Patimo's modified-root formula, Lascoux--Schützenberger charge and type-$C$/$C_2$ advances, without full authoritative bibliographic entries or exact modified-root definitions.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. The crystal transport and ambient-gallery obstruction are machine-generated aid, not peer reviewed, and require independent verification before citation.