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Fence Posets, Good Gradings and Frobenius Maximal Parabolics — arXiv:2609.10869

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.10869
License (as archived): CC0-1.0
Archived: 2026-10-16T03:49:27.692Z
SHA-256 of the archived text: 0cc5144fb7eed66889775d327d49c7fbfdbd4648fad735bdc2dff2730480c261
Stable URL of this page: https://metraipolis.com/documento/dc_cdff6e57eed1

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Fence Posets, Good Gradings and Frobenius Maximal Parabolics — arXiv:2609.10869 — arXiv (en), https://arxiv.org/abs/2609.10869 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cdff6e57eed1 (Metraipolis archive dc_cdff6e57eed1, sha256 0cc5144fb7eed66889775d327d49c7fbfdbd4648fad735bdc2dff2730480c261)

BibTeX

@misc{dccdff6e57eed1,
  title = {Fence Posets, Good Gradings and Frobenius Maximal Parabolics — arXiv:2609.10869},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.10869},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cdff6e57eed1; sha256 0cc5144fb7eed66889775d327d49c7fbfdbd4648fad735bdc2dff2730480c261},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.10869. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ 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 $\mathfrak{gl}_n$ 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 $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-14 in math.CO).
Categorie arXiv: Representation Theory (math.RT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.10869 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T03:18:59.226Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-14?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).