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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2406.10726 |
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| _version_ | 1866911093379563520 |
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| author | Stekolshchik, Rafael |
| author_facet | Stekolshchik, Rafael |
| contents | We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $Γ'$. The main question of the paper: what root $γ\in \varPhi$ can be added to $S'$ so that $S' \cup γ$ is also a subset of linearly independent roots? This extra root $γ$ is called the linkage root. The vector $γ^{\nabla}$ of inner products $\{(γ,τ'_i)\mid τ'_i \in S'\}$ is called the linkage label vector. Let $B_{Γ'}$ be the Cartan matrix associated with $Γ'$. It is shown that $γ$ is a linkage root if and only if $\mathscr{B}^{\vee}_{Γ'}(γ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{Γ'}$ is a quadratic form with the matrix inverse to $B_{Γ'}$. The set of all linkage roots for $Γ'$ is called a linkage system and is denoted by $\mathscr{L}(Γ')$. The Cartan matrix associated with any Carter diagram $Γ'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $Γ$, [St23]. The sizes of $\mathscr{L}(Γ')$ and $\mathscr{L}(Γ)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{Γ'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integer quadratic forms and extensions of subsets of linearly independent roots Stekolshchik, Rafael Representation Theory 17B22, 20F55, 15A63 We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $Γ'$. The main question of the paper: what root $γ\in \varPhi$ can be added to $S'$ so that $S' \cup γ$ is also a subset of linearly independent roots? This extra root $γ$ is called the linkage root. The vector $γ^{\nabla}$ of inner products $\{(γ,τ'_i)\mid τ'_i \in S'\}$ is called the linkage label vector. Let $B_{Γ'}$ be the Cartan matrix associated with $Γ'$. It is shown that $γ$ is a linkage root if and only if $\mathscr{B}^{\vee}_{Γ'}(γ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{Γ'}$ is a quadratic form with the matrix inverse to $B_{Γ'}$. The set of all linkage roots for $Γ'$ is called a linkage system and is denoted by $\mathscr{L}(Γ')$. The Cartan matrix associated with any Carter diagram $Γ'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $Γ$, [St23]. The sizes of $\mathscr{L}(Γ')$ and $\mathscr{L}(Γ)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{Γ'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented. |
| title | Integer quadratic forms and extensions of subsets of linearly independent roots |
| topic | Representation Theory 17B22, 20F55, 15A63 |
| url | https://arxiv.org/abs/2406.10726 |