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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.01956 |
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| _version_ | 1866909188624482304 |
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| author | Pan, Yang |
| author_facet | Pan, Yang |
| contents | Let $\mfp(d)$ be a standard parabolic subalgebra of $\mfsl_{n+1}(K)$ and $\mfu$ be the corresponding nilradical defined over an algebraically closed field $K$ of characteristic $p>0$. We construct a finite connected quiver $Q(d)$, through which we provide a combinatorial characterization of the centralizer $c_{\mfu}(x(d))$ of the Richardson element $x(d)$. We specifically focus on the centralizer when the levi factor of $\mfp(d)$ is determined by either one or two simple roots. This allows us to demonstrate that, under certain mild restrictions, the saturation rank of $\mfu$ equals the semisimple rank of the algebraic $K$-group $\SL_{n+1}(K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Saturation rank for nilradical of parabolic subalgebras in Type A Pan, Yang Representation Theory Let $\mfp(d)$ be a standard parabolic subalgebra of $\mfsl_{n+1}(K)$ and $\mfu$ be the corresponding nilradical defined over an algebraically closed field $K$ of characteristic $p>0$. We construct a finite connected quiver $Q(d)$, through which we provide a combinatorial characterization of the centralizer $c_{\mfu}(x(d))$ of the Richardson element $x(d)$. We specifically focus on the centralizer when the levi factor of $\mfp(d)$ is determined by either one or two simple roots. This allows us to demonstrate that, under certain mild restrictions, the saturation rank of $\mfu$ equals the semisimple rank of the algebraic $K$-group $\SL_{n+1}(K)$. |
| title | Saturation rank for nilradical of parabolic subalgebras in Type A |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2405.01956 |