Some Lagrangian quiver Grassmannians for the equioriented cycle
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914080040681472 |
|---|---|
| author | Micheli, Matteo |
| author_facet | Micheli, Matteo |
| contents | The goal of this paper is to better understand a family of linear degenerations of the classical Lagrangian Grassmannians $Λ(2n)$. It is the special case for $k=n$ of the varieties $X(k,2n)^{sp}$, introduced in previous joint work with Evgeny Feigin, Martina Lanini and Alexander Pütz. These varieties are obtained as isotropic subvarieties of a family of quiver Grassmannians $X(n,2n)$, and are acted on by a linear degeneration of the algebraic group $Sp_{2n}$. We prove a conjecture proposed in the paper above for this particular case, which states that the ordering on the set of orbits in $X(n,2n)^{sp}$ given by closure-inclusion coincides with a combinatorially defined order on what are called symplectic $(n,2n)$-juggling patterns, much in the same way that the $Sp_{2n}$ orbits in $Λ(2n)$ are parametrized by a type C Weyl group with the Bruhat order. The dimension of such orbits is computed via the combinatorics of bounded affine permutations, and it coincides with the length of some permutation in a Coxeter group of affine type C. Furthermore, the varieties $X(n,2n)$ are GKM, that is, they have trivial cohomology in odd degree and are equipped with the action of an algebraic torus with finitely many fixed points and 1-dimensional orbits. In this paper it is proven that $X(n,2n)^{sp}$ is also GKM, with respect to the action of a subtorus of the above torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Lagrangian quiver Grassmannians for the equioriented cycle Micheli, Matteo Representation Theory Algebraic Geometry Combinatorics 16G20 (Primary) 14M15 (Secondary) The goal of this paper is to better understand a family of linear degenerations of the classical Lagrangian Grassmannians $Λ(2n)$. It is the special case for $k=n$ of the varieties $X(k,2n)^{sp}$, introduced in previous joint work with Evgeny Feigin, Martina Lanini and Alexander Pütz. These varieties are obtained as isotropic subvarieties of a family of quiver Grassmannians $X(n,2n)$, and are acted on by a linear degeneration of the algebraic group $Sp_{2n}$. We prove a conjecture proposed in the paper above for this particular case, which states that the ordering on the set of orbits in $X(n,2n)^{sp}$ given by closure-inclusion coincides with a combinatorially defined order on what are called symplectic $(n,2n)$-juggling patterns, much in the same way that the $Sp_{2n}$ orbits in $Λ(2n)$ are parametrized by a type C Weyl group with the Bruhat order. The dimension of such orbits is computed via the combinatorics of bounded affine permutations, and it coincides with the length of some permutation in a Coxeter group of affine type C. Furthermore, the varieties $X(n,2n)$ are GKM, that is, they have trivial cohomology in odd degree and are equipped with the action of an algebraic torus with finitely many fixed points and 1-dimensional orbits. In this paper it is proven that $X(n,2n)^{sp}$ is also GKM, with respect to the action of a subtorus of the above torus. |
| title | Some Lagrangian quiver Grassmannians for the equioriented cycle |
| topic | Representation Theory Algebraic Geometry Combinatorics 16G20 (Primary) 14M15 (Secondary) |
| url | https://arxiv.org/abs/2507.15841 |