Fence Posets, Good Gradings and Frobenius Maximal Parabolics
Abstract
Let L be a Frobenius maximal parabolic subalgebra of sln. For any F∈L* for which the Kirillov form BF(x,y)=F([x,y]) is non-degenerate, let F̂ denote the associated principal element. We prove that the multiplicities of the eigenvalues of adF̂ on L form a unimodal sequence symmetric about 1/2. We also prove that the multiplicities of the eigenvalues of adF̂ on gln form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to L to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of gln in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element e lies in L. The two unimodality results then follow from the injectivity properties of ade coming from the good grading and the duality induced by the bilinear form BF.
Related
- Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity
- Combinatorial interpretation of the coefficients of the order polynomial of fence posets
- Maximal anti-Ramsey problems for posets
- Symmetry breaking in the polygonal Szegö-Weinberger inequality as p→1+: the longest shortest-fence quadrilateral
- The fence conjecture for convex projective polygons
BibTeX
@misc{giaquinto2026fence,
title = {{Fence Posets, Good Gradings and Frobenius Maximal Parabolics}},
author = {Anthony Giaquinto and John Irving and Aaron Lauve and Mitja Mastnak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10869},
eprint = {2609.10869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10869},
}
RIS
TY - UNPB
TI - Fence Posets, Good Gradings and Frobenius Maximal Parabolics
AU - Anthony Giaquinto
AU - John Irving
AU - Aaron Lauve
AU - Mitja Mastnak
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10869
UR - https://arxiv.org/abs/2609.10869
AB - Let L be a Frobenius maximal parabolic subalgebra of sl_n. For any F∈L^* for which the Kirillov form B_F(x,y)=F([x,y]) is non-degenerate, let F̂ denote the associated principal element. We prove that the multiplicities of the eigenvalues of ad_F̂ on L form a unimodal sequence symmetric about 1/2. We also prove that the multiplicities of the eigenvalues of ad_F̂ on gl_n form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to L to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of gl_n in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element e lies in L. The two unimodality results then follow from the injectivity properties of ad_e coming from the good grading and the duality induced by the bilinear form B_F.
ER -