The Clifford-cyclotomic group and Euler–Poincaré characteristics

Abstract

AbstractFor 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 $\operatorname {\mathrm {SO_{3}}}(R_n^+)$ .