The Clifford-cyclotomic group and Euler–Poincaré characteristics
| Autor: | Bruce W. Jordan, Allan Keeton, Yevgeny Zaytman, Colin Ingalls, Adam Logan |
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| Rok vydání: | 2020 |
| Předmět: |
Discrete mathematics
Group (mathematics) General Mathematics 010102 general mathematics Unitary matrix 01 natural sciences Set (abstract data type) symbols.namesake Integer If and only if 0103 physical sciences Poincaré conjecture Euler's formula symbols 0101 mathematics 010306 general physics Hadamard matrix Mathematics |
| Zdroj: | Canadian Mathematical Bulletin. 64:651-666 |
| ISSN: | 1496-4287 0008-4395 |
| DOI: | 10.4153/s0008439520000727 |
| Popis: | For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$ . Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$ . Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$ and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$ . We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$ . We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$ , $\operatorname {\mathrm {PSU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_2^\zeta }}(R_n)$ , and $\operatorname {\mathrm {SO_{3}}}(R_n^+)$ . |
| Databáze: | OpenAIRE |
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