The Clifford-cyclotomic group and Euler-Poincaré characteristics

Abstract

For an integer $n\ge 8$ divisible by $4$, let $R_n=\mathbb{Z}[{\zeta }_n,1/2]$ and let $U_2(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $U_2^{\zeta }(R_n)=\{\gamma \in U_2(R_n)\mid \det \gamma \in ⟨{\zeta }_n⟩\}$. Let ${\mathcal{G}}_n\subseteq U_2^{\zeta }(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac{1}{2}[\begin{array}{cc}1+i& 1+i\\ 1+i& -1-i\end{array}]$ and the gate $T=[\begin{array}{cc}1& 0\\ 0& {\zeta }_n\end{array}]$. We prove that ${\mathcal{G}}_n=U_2^{\zeta }(R_n)$ if and only if $n=8,12,16,24$ and that $[U_2^{\zeta }(R_n):{\mathcal{G}}_n]=\infty$ if $U_2^{\zeta }(R_n)\ne {\mathcal{G}}_n$. We compute the Euler-Poincaré characteristics of the groups ${\mathrm{SU}}_2(R_n)$, ${\mathrm{PSU}}_2(R_n)$, ${\mathrm{PU}}_2(R_n)$, ${\mathrm{PU}}_2^{\zeta }(R_n)$, and ${\mathrm{SO}}_3(R_n^{+})$.