2019/03/22 by Ingalls, Colin J., Jordan, Bruce W., Keeton, Allan +2 · 1 citation
#20G30 (Secondary) #81P45 (Primary) #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1903.09497
For an integer n≥ 8 divisible by 4, let Rn=ℤ[ζn,1/2] and let U2(Rn) be the group of 2× 2 unitary matrices with entries in Rn. Set U2ζ(Rn)=\γ\inU2(Rn)| detγ∈⟨ζn⟩\. Let Gn⊆ U2ζ(Rn) be the Clifford-cyclotomic group generated by a Hadamard matrix H=(1)/(2)[\beginsmallmatrix 1+i & 1+i 1+i &-1-i\endsmallmatrix] and the gate T=[\beginsmallmatrix1 & 0 0 & ζn\endsmallmatrix]. We prove that Gn=U2ζ(Rn) if and only if n=8, 12, 16, 24 and that [U2ζ(Rn):Gn]=∞ if U2ζ(Rn)≠ Gn. We compute the Euler-Poincaré characteristics of the groups SU2(Rn), PSU2(Rn), PU2(Rn), PUζ2(Rn), and SO3(Rn+).