A mathematical preprint describes a systematic way to classify the basic building blocks of equal-rank rank-two Lusztig-Vogan categories and track the actions of generating Soergel-bimodules on them. It also proposes an algorithm for arbitrary finite-rank categories and reports tests on low-rank examples.
At the center are indecomposable objects, the summands that remain after the relevant decompositions. The study asks how to classify them, decide when two are isomorphic, and describe the actions of the generating Soergel-bimodules. It uses W-graphs, diagrams of the resulting object-and-action structure, to present that output. The paper also states that its category categorifies the principal block of the corresponding Lusztig-Vogan module, meaning that the category is used to represent that module-level structure through objects and actions.
Making the algebra computable
The hand-classification recipe starts with generators. It acts on them by Bott-Samelson bimodules, decomposes the results into indecomposable summands, checks which summands are isomorphic and repeats. In practical terms, each pass asks whether an operation has produced a new building block or one already found.
The computational version stores the degrees of a graded right-R basis and a graded right-R endomorphism for each generator of the invariant ring. To find direct summands, it uses primitive idempotents, whose images are indecomposable. These idempotents serve as algebraic projectors that isolate the pieces the algorithm is looking for.
Four cases, different outputs
The worked equal-rank rank-two cases are SU(2, 1), Sp(1, 1), Sp4 (R) and split G2. In the type A example, the displayed graph has nodes named Bbig, RBt, Rt Bs, R, Rt and Rts, each identified as an indecomposable object.
The type C example's displayed graph has Bbig, RBt, R and Rt as its indecomposable nodes.
For split G2, the implementation displays the resulting W-graph. The Sp4 (R) implementation returns the reported W-graph.
Tests and practical limits
The reported tests include all categories of ranks 1, 2 and 3, as well as the rank 4 form SU(4, 1) and rank 5 form SU(5, 1). In real-reductive-group examples, the paper reports that the algorithm returned the corresponding correct W-graph.
The paper identifies a clear computational bottleneck. Gröbner-basis computation during primary decomposition is expected to slow the procedure substantially and consume large amounts of memory for larger examples. The paper therefore flags larger examples as a practical challenge.
The manual G2 calculation has its own unresolved point: the available tools did not establish that B1 and B2 were indecomposable.
From theorem to implementation
Taken together, the preprint presents classification, category actions and graph recovery as linked parts of one computational program. It reports a Magma implementation and says that source code can be found at the cited DRZ26 repository. The document is an arXiv preprint, version v1, dated 28 Aug 2026.
Its acknowledgements report scholarship support for the first and third authors, a Max Planck Institute acknowledgement for the second author, and use of ChatGPT 5.6 for proofreading and figure formatting. They attribute all writing and mathematical arguments to the authors.
Paper data and sources
Original title: Lusztig-Vogan categories of equal rank 2
Authors: Daniel Dunmore, Anna Romanov, Victor L. Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text