Charge, the Brylinski--Kostant filtration, and canonical bases

Definitions and mathematical statement

The source asks for a representation-theoretic explanation of charge and a relation between filtration-adapted parametrizations and Lusztig--Kashiwara canonical bases. It does not specify a representation, weight, filtration, or meaning of "non-miraculous." The aid uses the type-$A$ Brylinski--Kostant (BK) filtration and distinguishes equality of graded dimensions from a canonical filtered-basis identification.

Let $V_\lambda$ be a polynomial irreducible $\mathrm{GL}_n({\mathbb{C}})$ representation, $\mu$ a dominant weight of the same degree, and $e$ a chosen principal raising nilpotent. On $V=V_\lambda[\mu]$ define \[ F_i=V\cap\ker(e^{i+1}),\qquad F_{-1}=0. \] Let $E$ have formal basis indexed by semistandard tableaux $T$ of shape $\lambda$ and content $\mu$, and let $C_i$ be the span of basis elements with $\operatorname{charge}(T)\le i$. With the appropriate dominant-weight and Kostka--Foulkes conventions, the aid invokes \[ \sum_i\dim(F_i/F_{i-1})t^i =K_{\lambda\mu}(t) =\sum_Tt^{\operatorname{charge}(T)}. \] Thus the two flags have the same dimension vector, but this does not say that the canonical basis itself is adapted to $F$.

The reported exact reduction considers \[ \mathcal T(C,F)=\{g:E\xrightarrow{\sim}V:g(C_i)=F_i\text{ for every }i\}. \] It is nonempty. Let $P_C$ and $P_F$ be the parabolic stabilizers of the respective flags, omitting repeated steps. Left composition by $P_F$ and right composition by $P_C$ make it a transporter bitorsor, each action free and transitive. Existence of many such isomorphisms is automatic from the dimension equality; the missing geometric/canonical datum is a distinguished point or otherwise justified choice, not another equality of Hilbert series.

For $V_\lambda=\operatorname{Sym}^d({\mathbb{C}}^n)$ and dominant $\mu=(\mu_1,\ldots,\mu_n)$, $\sum_j\mu_j=d$, the weight space is one-dimensional. In polynomial coordinates take $e=\sum_{j=1}^{n-1}x_j\partial_{x_{j+1}}$. The aid reports that its unique global-basis line first appears in BK degree \[ n(\mu)=\sum_j(j-1)\mu_j, \] matching the charge of the unique one-row tableau of that content. The global-basis vector is proportional to the weight monomial; its normalization is not fixed by this filtration statement. In this case the transporter ambiguity reduces to a nonzero scalar.

The aid also attributes to Patimo (2025) an affine-Grassmannian geometric/crystal explanation of type-$A$ charge not based on cyclage. It says lifting this to MV cycles and distinguished filtered bases remains open in the literature checked. Full hypotheses and definitions are absent, so neither a canonical comparison in general nor extension to other types is asserted.

Short English statement

Charge and the BK filtration have matching graded dimensions in the supplied type-$A$ setting. The aid identifies the extra choice needed to compare actual filtered bases and checks the one-row case exactly. Its cited geometric explanation does not yet supply the requested general canonical-basis lift.

Sources

• Local UnsolvedMath registry, permanent ID 20002911, source code catalogue entry; created and updated 14 August 2026. Canonical aim-special-functions-notes.json, zero-based notes[15], tag problem, item 16.
• AIM "Generalized Kostka polynomials": https://aimath.org/WWN/kostka/prob2.pdf. Wording is marked exact; original remarks and literature field are empty. Lusztig--Kashiwara bases are named without a complete bibliographic reference.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. It invokes BK/Kostka results and Patimo (2025), but supplies no full titles, theorem statements, or authoritative bibliographic links.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.