Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings
| Autor: | Allan Keeton, Colin Ingalls, Yevgeny Zaytman, Bruce W. Jordan, Adam Logan |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Number Theory
Group (mathematics) General Mathematics Group Theory (math.GR) Unitary state Prime (order theory) Combinatorics Tree (descriptive set theory) Mathematics::Group Theory FOS: Mathematics Number Theory (math.NT) Amalgam (chemistry) Mathematics::Representation Theory Primary 20G30 Secondary 11R18 81P45 Mathematics - Group Theory Quotient Mathematics |
| Popis: | Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(\zeta_n+\bar{\zeta}_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[\zeta_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[\zeta_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $\Delta =\Delta_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8\leq n\leq 48$, $n\neq 44$, as well as for $\operatorname{PU_{2}}(\mathbb{Z}[\zeta_{60}, 1/2])$. |
| Databáze: | OpenAIRE |
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