A three-way agreement
A mathematical preprint reports that three ways of grouping elements of a canonical basis—the coordinate, total-positivity and representation-theoretic partitions—are equal in the paper’s stated setting. The result is given for every Weyl-group element covered by that setting. The authors interpret the combined identifications as a proof of Lusztig’s conjecture, which concerns agreement between the relevant Lusztig and Kashiwara constructions.
Here, a partition simply means a way of grouping the basis elements. The paper shows that the groups produced by these routes have the same membership in the stated setting.
That conclusion is conditional on the framework used. The work assumes throughout that G is of simply-laced type, so the equality is presented within that class and not beyond the paper’s stated setting.
How the comparison is built
To make the comparison, the analysis combines a tropical parametrization of the canonical basis with Kashiwara crystal operators acting on that basis. It also uses Demazure modules generated by extremal canonical-basis vectors. Together, these methods connect the coordinate, total-positivity and representation-theoretic descriptions that the paper proves equal.
A geometric identity
The paper then moves from partitions of the basis to a geometric setting called an open Richardson variety. For a regular weight λ and Weyl-group elements v and w with v≤w, it states that the total-positivity subset attached to that variety equals the intersection of the corresponding Demazure and opposite-Demazure subsets. In plain terms, membership in the geometric subset is captured by satisfying both representation-theoretic conditions at once.
A second identification handles the opposite-Demazure side. The canonical basis of each opposite Demazure module is identified with the total-positivity subset indexed by the interval ending at w0. This supplies the link needed to express the open-Richardson subset as an intersection, tying the geometric and representation-theoretic descriptions together.
The type-A consequence
In type A, the paper pushes the connection into a weight test. It gives an if-and-only-if characterization: a weight belongs to the relevant set exactly when it satisfies the root-lattice condition and lies in the corresponding Bruhat interval polytope. The two conditions are therefore jointly necessary and sufficient within that theorem.
The proof uses an integer-scaling criterion for the polytope. Membership is equivalent to the existence of an integer scaling that produces a weight in the corresponding total-positivity subset. It also invokes a line-intersection lemma, which supplies a point in the smaller Bruhat interval polytope while preserving the root-lattice condition. The saturation proof is explicitly organized into three cases.
Where the claim stops
The boundaries are part of the result. The partition equality is stated under the simply-laced assumption; the Richardson compatibility statement requires regular λ and v≤w; and the saturation characterization is stated in type A. The supplied claims therefore do not establish these same conclusions outside those conditions.
The document is labeled as an arXiv version-one preprint in the math.RT category. Its findings are presented within that mathematical framework and its stated assumptions.
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