A generic clique-block basis of individual LLT polynomials

Definitions and mathematical statement

The source asks for other interesting symmetric-function bases simply related to LLT polynomials. It is exploratory, not a single yes/no conjecture. Its historical remarks say that Schur functions occur among LLTs; Lam includes Hall--Littlewood and skew Schur functions; Morse gives $s^{(3)}_{(2,1,1)}$ as a purported non-LLT $k$-Schur example; and modified Macdonald polynomials, and conjecturally $\nabla(e_n)$, have weighted LLT expansions using ribbon tuples and shifted-column tuples. These remarks suppress normalization and definition choices and are not fresh status assertions.

The aid constructs an explicit family over ${\mathbb{Q}}(q)$. For an ordered complete attacking block of $r$ unicellular LLT cells, use inversion normalization \[ C_r(X;q)=\sum_{i_1,\ldots,i_r\ge1} q^{\#\{a<b:i_a>i_b\}}x_{i_1}\cdots x_{i_r},\qquad C_0=1. \] For a partition $\lambda$, let $C_\lambda=\prod_{r\in\lambda}C_r$, treating repeated parts with multiplicity. This product is itself the unicellular LLT for an ordered disjoint union of clique blocks, with no attacks between blocks. The choice of inversion normalization is essential to its specialization statements.

For every degree $n$, the aid reports that $\{C_\lambda:\lambda\vdash n\}$ is a basis of $\Lambda_n\otimes{\mathbb{Q}}(q)$. With matching partition order for rows and columns, its transition determinant to the complete basis $h_\lambda=\prod_{r\in\lambda}h_r$ is \[ D_n(q)=\prod_{\lambda\vdash n}\prod_{r\in\lambda} \prod_{j=1}^{r-1}(1-q^j). \] Hence at a complex specialization $q=a$ the degree-$n$ family is a basis exactly when $a^j\ne1$ for $1\le j\le n-1$. The condition is empty in degrees zero and one. At $q=0$ it is the complete basis, while at $q=1$ every degree-$n$ member becomes $h_1^n$; this is a rank-one collapse when $n\ge2$. Over all degrees simultaneously, a fixed root of unity eventually causes failure, whereas a non-root-of-unity specialization avoids these determinant zeros.

This supplies one generic basis of individual LLTs, not a classification of all interesting bases or a claim that every $k$-Schur function is an LLT. The aid flags the historical Morse example as convention-dependent and inconsistent with a later bandwidth-three realization attributed to Miller. Neither the exact $k$-Schur/LLT convention nor that later theorem is supplied, so the conflicting example is retained as a reference/normalization issue rather than silently resolved.

Short English statement

The aid gives a partition-indexed basis of individual clique-block LLTs at generic parameter, with an exact determinant describing every bad numerical specialization. It also records that the source's historical $k$-Schur example depends on conventions and later literature not fully supplied here.

Sources

• Local UnsolvedMath registry, permanent ID 20002904, source code catalogue entry; created and updated 14 August 2026. Canonical aim-special-functions-notes.json, zero-based notes[8], tag problem, item 9.
• AIM "Generalized Kostka polynomials": https://aimath.org/WWN/kostka/prob2.pdf. The supplied verification reports no visible OCR error; wording is marked exact. Original remarks and literature field are empty. Lam and Morse are named in the statement, without complete bibliographic references.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. It names Miller's later bandwidth-three realization but provides no full title, date, theorem statement, or authoritative reference; the determinant/basis claim is likewise supplied without a full citation.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.