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| Auteur principal: | |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2406.16179 |
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| _version_ | 1866914846841241600 |
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| author | Aslan, Songul |
| author_facet | Aslan, Songul |
| contents | Let $\mathscr{X}$ be the affine flag manifold of Lie type $A_{n-1}^{(1)}$ where $n \geq 3$ and let $W_{\text{aff}}$ be the associated affine Weyl group. The moment graph for $\mathscr{X}$ encodes the torus fixed points (corresponding to elements of the affine Weyl group $W_{\text{aff}}$) and the torus stable curves in $\mathscr{X}$. Given a fixed point $u\in W_{\text{aff}}$ and a degree $\mathbf{d}=(d_0,d_1,...,d_{n-1})\in \mathbb{Z}_{\geq 0}^{n}$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $\mathscr{X}$ which can be reached from $u'\leq u$ by a chain of curves of total degree $\leq \mathbf{d}$. In this paper we give combinatorial formulas and algorithms for calculating these elements in $\mathscr{X}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorial Curve Neighborhood of the Affine Flag Manifold of Type $A_{n-1}^1$ Aslan, Songul Combinatorics Let $\mathscr{X}$ be the affine flag manifold of Lie type $A_{n-1}^{(1)}$ where $n \geq 3$ and let $W_{\text{aff}}$ be the associated affine Weyl group. The moment graph for $\mathscr{X}$ encodes the torus fixed points (corresponding to elements of the affine Weyl group $W_{\text{aff}}$) and the torus stable curves in $\mathscr{X}$. Given a fixed point $u\in W_{\text{aff}}$ and a degree $\mathbf{d}=(d_0,d_1,...,d_{n-1})\in \mathbb{Z}_{\geq 0}^{n}$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $\mathscr{X}$ which can be reached from $u'\leq u$ by a chain of curves of total degree $\leq \mathbf{d}$. In this paper we give combinatorial formulas and algorithms for calculating these elements in $\mathscr{X}$. |
| title | Combinatorial Curve Neighborhood of the Affine Flag Manifold of Type $A_{n-1}^1$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.16179 |