Classification of the root systems $R(m)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910943009570816 |
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| author | Polo, Patrick |
| author_facet | Polo, Patrick |
| contents | Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple of $m$. In a recent paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems $R(m)$; in particular, they introduced a constant $d_m$ which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to $R(m)$ and equals $1$ if the roots of height $m$ form a base of $R(m)$, and proved this property when $R$ is of type $A$ or $C$, and also in type $B$ if $m$ is odd. In this paper, we complete their analysis by determining a base of $R(m)$ and computing the constant $d_m$ in all cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of the root systems $R(m)$ Polo, Patrick Representation Theory 17B22, 22E46 Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple of $m$. In a recent paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems $R(m)$; in particular, they introduced a constant $d_m$ which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to $R(m)$ and equals $1$ if the roots of height $m$ form a base of $R(m)$, and proved this property when $R$ is of type $A$ or $C$, and also in type $B$ if $m$ is odd. In this paper, we complete their analysis by determining a base of $R(m)$ and computing the constant $d_m$ in all cases. |
| title | Classification of the root systems $R(m)$ |
| topic | Representation Theory 17B22, 22E46 |
| url | https://arxiv.org/abs/2504.09204 |