A Garsia--Haiman filtration theorem and a small-k obstruction for diagonal harmonics

Definitions and mathematical statement

Let $\Lambda$ be the symmetric-function ring over ${\mathbb{Q}}$, and let $\widetilde H_\mu(X;q,t)$ be the modified Macdonald functions. Define $n(\mu)=\sum_{i\ge1}(i-1)\mu_i$, and define $\nabla$ by \[ \nabla\widetilde H_\mu=q^{n(\mu')}t^{n(\mu)}\widetilde H_\mu. \] The source identifies the bigraded Frobenius series of diagonal harmonics of degree $n$ with $\nabla e_n$, and that of the Garsia--Haiman module indexed by $\mu\vdash n$ with $\widetilde H_\mu$. For a bigraded $S_n$-module $M$, this series is $\sum_{a,b}q^at^b\operatorname{ch}(M_{a,b})$, where $\operatorname{ch}$ is the Frobenius characteristic. The problem asks for meaningful relationships between these modules or series and $k$-Schur functions, for positive integers $k$. It is a research question rather than a single asserted identity or filtration.

For the Garsia--Haiman part, the supplied aid cites Kato's Corollary 10.4: for each $k$-bounded partition $\mu$, meaning $\mu_1\le k$, a specified pullback of the Garsia--Haiman module has a genuine filtration by Chen--Haiman modules whose characters are modern $q$-parameter $k$-Schur functions. The precise pullback, module category, grading and character conventions are not supplied here. In particular, this must not be read as an unqualified filtration of the original bigraded $S_n$-module, or as an identification of every historical $k$-Schur parameter convention. The aid reports refined Macdonald positivity in that specified convention.

For diagonal harmonics the aid reports, for $n\ge2$, \[ [h_n]\left((\nabla e_n)|_{q=t=1}\right)=(-1)^{n-1}. \] Here $[h_n]$ means the coefficient of the complete-basis element $h_{(n)}$ in the expansion in the basis $\{h_\lambda\}_{\lambda\vdash n}$, not a Schur coefficient and not an unspecified scalar product. If $k<n$, every element of $\Lambda_{(k)}={\mathbb{Q}}[h_1,\ldots,h_k]$ has zero such coefficient in degree $n$. Consequently there can be no finite direct identity \[ \nabla e_n=\sum_{\substack{\lambda\vdash n\\\lambda_1\le k}}a_\lambda(q,t)\,B_\lambda^{(k)}(X;q,t) \] when each coefficient is regular at $(1,1)$, each basis function is regular there, and $B_\lambda^{(k)}(X;1,1)\in\Lambda_{(k)}$, as for the $k$-Schur convention in the reported obstruction. Regular means having a well-defined finite specialization at that point. This excludes positive polynomial direct expansions and character-compatible finite filtrations with those characters. It does not exclude singular-coefficient identities, projections, derived constructions or other indirect relationships. For $k\ge n$, the aid records the stable Schur regime and ordinary Schur positivity as the stable-range relationship, subject to the stated parameter convention. These observations do not resolve every possible meaning of "nice relationships.

Short English statement

Seek module-theoretic or symmetric-function links between $k$-Schur functions, diagonal harmonics and Garsia--Haiman modules. The supplied aid reports a convention-dependent pullback filtration theorem for the latter and an obstruction to regular direct small-$k$ expansions of the former, while leaving other relationships possible.

Sources

• AIM workshop Generalized Kostka polynomials, Problem (14), https://aimath.org/WWN/kostka/prob2.pdf.
• The research summary cites Kato, Corollary 10.4, and names Chen--Haiman modules; it gives no full publication title, date, URL or complete module and pullback definitions.
• Local UnsolvedMath registry, permanent ID 20002909, exact code catalogue entry; canonical location aim-special-functions-notes.json, notes[13], dated 14 August 2026; statement labeled exact.
• Research attempt 1: partial result, problem partially solved at the run, rigor label proved, novelty confidence low; registry status partially solved. The filtration attribution, obstruction and stable-range account are supplied machine-generated research aids, not peer reviewed, requiring independent verification before citation.