One dimension across a fixed sheet
A mathematical preprint reports that type A Hessenberg varieties keep the same dimension when their defining matrix changes within a fixed sheet, provided the Hessenberg function stays fixed. In this setting, the paper groups matrices into sheets indexed by partitions. The result applies across every partition and Hessenberg function covered by the paper.
Hessenberg varieties are spaces of flags defined by a matrix X and a Hessenberg function h. The paper asks how the dimension of the resulting space changes as X varies while h is held constant. It studies a parametrized family of algebraic varieties and matrices, rather than an empirical sample or a statistical data set.
The authors compare two special kinds of matrices associated with the same sheet. For a fixed h and partition, the Hessenberg variety attached to a nilpotent representative has the same dimension as the one attached to a semisimple representative. They also show that, for a matrix meeting the paper’s lambda-standard condition, the associated Hessenberg variety has the same dimension as a semisimple variety in that sheet.
The argument works through affine cells
The proof uses affine pavings, a way of breaking each variety into affine cells. The dimension comparison is then reduced to comparing the largest cell dimensions in the nilpotent and semisimple varieties. The largest cells are the pieces that can determine the dimension of the full variety.
A second step concerns whether a Hessenberg-Schubert cell is nonempty, meaning that the cell actually appears in the paving. After the decomposition used in the proof, a cell is nonempty for X exactly when the corresponding cell is nonempty for the nilpotent component. That equivalence allows the authors to compare the two varieties through their nilpotent-side cells.
At the cell level, the comparison is expressed using tableaux and a condition called h-strictness. For h-strict tableaux, the nilpotent cell has dimension at least as large as the corresponding semisimple cell. The two dimensions are equal when the tableau is column increasing.
The construction for a lambda-standard matrix supplies the link to a same-sheet semisimple matrix. Together with the equal-dimension result for nilpotent and semisimple representatives, it leads to the paper’s sheet-wide conclusion: every matrix in a fixed sheet gives a Hessenberg variety with the same dimension. That common value is the dimension of a nilpotent Hessenberg variety from the sheet’s unique nilpotent orbit.
What the result does not settle
The main theorem is about dimension, a single numerical measure of the spaces. It does not establish that all the varieties in a sheet are isomorphic or have identical topology. The common dimension is identified through a nilpotent reference variety, but the supplied analysis does not give a general closed formula for arbitrary partitions and Hessenberg functions.
The paper also advances a conjecture about the Poincaré polynomial of Hess(X,h). It proposes that this polynomial dominates the one for the nilpotent variety associated with the same sheet when X lies in that sheet. The statement is presented as an unproved conjecture and is separate from the dimension theorem.
A broader comparison remains open
The document reports confirmation of the conjecture for the semisimple choice X = Sλt and for every partition λ when n is at most 7. That bounded result does not establish the conjecture in full generality. A geometric or representation-theoretic explanation for the dimension phenomenon remains an open direction identified in the analysis.
The paper is still a preprint
The work is an arXiv preprint, version 1, dated 25 Aug 2026. Its funding statement reports partial support for the first author from an NSERC Discovery Grant and a Canada Research Chair Tier I Award, support for the second author from an NSF CAREER grant, and support for the third author from an NSF MSPRF award.
Paper data and sources
Original title: Dimensions of type $A$ Hessenberg varieties over a fixed sheet
Authors: Megumi Harada, Martha Precup, Colleen Robichaux
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text