Maximal root subsystems of affine reflection systems and duality
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912783789981696 |
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| author | Habib, Irfan |
| author_facet | Habib, Irfan |
| contents | Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an affine reflection system (reduced and non-reduced) and prove that this result holds in much more generality for reduced affine reflection systems. Moreover, we explicitly determine when a maximal root subsystem is a maximal closed root subsystem. Using our classification, at the end, we characterize the maximal root systems of affine reflection systems with nullity less than or equal to $2$ using Hermite normal forms; especially for Saito's EARS of nullity $2.$ This in turn classifies the maximal subgroups of the Weyl group of an affine reflection system that are generated by reflections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal root subsystems of affine reflection systems and duality Habib, Irfan Rings and Algebras Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an affine reflection system (reduced and non-reduced) and prove that this result holds in much more generality for reduced affine reflection systems. Moreover, we explicitly determine when a maximal root subsystem is a maximal closed root subsystem. Using our classification, at the end, we characterize the maximal root systems of affine reflection systems with nullity less than or equal to $2$ using Hermite normal forms; especially for Saito's EARS of nullity $2.$ This in turn classifies the maximal subgroups of the Weyl group of an affine reflection system that are generated by reflections. |
| title | Maximal root subsystems of affine reflection systems and duality |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2411.08749 |