Three ways of partitioning a canonical basis turn out to give exactly the same result, according to a mathematical preprint. The author presents the equality as a proof of a conjecture of Lusztig. The finding is a theorem within a specified algebraic setting, not a result drawn from observations or experiments.
The paper studies canonical bases associated with quantum groups for a split connected reductive group over the real numbers, in the simply-laced case. It considers those bases alongside representations and strata in a flag variety.
A shared mathematical core
The central argument identifies all three partitions with the same subset of the canonical-basis parameter space. In other words, the agreement comes from a shared mathematical description of the objects being divided.
The paper also studies a subset defined by total positivity. It finds that this subset is independent of which point is chosen in the positive Richardson stratum.
The Richardson connection
Another theorem concerns open Richardson varieties and their canonical-basis subsets. For a regular highest weight and Weyl-group elements v and w with v at or below w in Bruhat order, the subset associated with the open Richardson variety equals the intersection of the corresponding ordinary and opposite Demazure subsets.
The result also has a one-sided counterpart. The canonical basis of an opposite Demazure module is identified with the total-positive subset whose upper endpoint is the longest element of the Weyl group. This identification supplies one of the two pieces in the Richardson intersection.
The scope is specific. The Richardson equality is stated for regular weights and ordered Weyl-group elements, while the supplied analysis says the singular-weight remark is restricted to minimal-length representatives. It is not presented as a theorem covering all singular cases.
Weights inside a polytope
The preprint then turns to weights and geometry. In type A, under the section's regular-weight assumption, a candidate weight belongs to the relevant Richardson canonical-basis weight set exactly when two conditions are met: its difference from the highest weight lies in the root lattice, meaning the lattice generated by the roots, and the candidate itself lies in the corresponding Bruhat interval polytope.
A Bruhat interval polytope is the geometric object attached here to the interval between two Weyl-group elements in Bruhat order. The result says that, together with the root-lattice condition, this geometric test captures precisely the weights represented in the canonical-basis subset.
The proof connects the polytope to the representation through scaling. A point belongs to the polytope exactly when some integer scaling factor N, with N at least 1, makes the scaled weight appear in the corresponding total-positive subset at highest weight scaled by the same factor. The proof strategy also uses induction on Weyl-group length.
No broader conclusion is supplied: the saturation result is limited to type A and the regular-weight setting described in the paper. The supplied analysis does not establish the result for other group types or for weights outside those assumptions.
A result with clear boundaries
These are formal algebraic claims, not empirical findings. The study has no human, animal, cell or other empirical sample; its objects of analysis are mathematical structures and basis subsets.
Within its definitions and assumptions, the paper states exact equalities and an exact membership criterion. It does not report measurements or statistical estimates.
The document is an arXiv preprint labeled arXiv:2608.24028v2 and dated 4 September 2026. The supplied text reports no funding source and acknowledges Huanchen Bao for discussions and suggestions.
Paper data and sources
Original title: Partitions of canonical bases
Authors: Jeff York Ye
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text