A178176 a(n) is the number of central quotients of simple compact Lie groups of dimension n.

0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Author

Andrew Rupinski, Dec 18 2010

Keywords

• nonn

Examples

			a(3) = 2 since the 3-dimensional SU(2) has two central quotients: SU(2) and SU(2)/2 = SO(3).
a(28) = 3 and not 4 since, because of triality for Spin(8), the semi-spin group HSpin(8) is isomorphic to SO(8). Thus, the only groups are Spin(8), SO(8), PSO(8). See the nLab link.
The unusually large value a(78) = 6 is due to Spin(13), SO(13), Sp(6), PSp(6), E_6, E_6/Z3 all of dimension 78.
		

Links

• Andrea Aveni, Table of n, a(n) for n = 1..1000
• Math Overflow, Number of compact connected Lie groups of given dimension
• nLab, semi-spin group

Programs

R

Number.Divisors=function(n){
  out=c()
  for(j in 1:n){if(n%%j==0){out=c(out,j)}}
  return(length(out))
}
a178176=function(n){
  kSU=sqrt(n+1)
  kSO=(sqrt(8*n+1)+1)/2
  kSp=(sqrt(8*n+1)-1)/4
  a=0
  if(n %in% c(14,52,248)){a=a+1} # G2, F4, E8 with center Z1
  if(n %in% c(78,133)){a=a+2} # E7 with center Z2, E6 with center Z3
  if(kSp%%1==0 & kSp>=2){a=a+2} # Sp(k), PSp(k)
  if(kSU%%1==0 & kSU>=2){a=a+Number.Divisors(kSU)} # SU(n)/Zd
  if(kSO%%1==0 & kSO>=7 & kSO!=8){
    if(kSO%%2!=0){a=a+2} # Spin(k), SO(k)
    if(kSO%%2==0 & kSO%%4!=0){a=a+3} # Spin(k), SO(k), PSO(k)
    if(kSO%%4==0){a=a+4} # Spin(k), SO(k), HSpin(k), PSO(k)
  }
  if(n==28){a=3} # Because of Triality: Spin(8), HSpin(8)=SO(8), PSO(8)
  return(a)
} # Andrea Aveni, Mar 23 2025

Extensions

a(28) corrected by Andrea Aveni, Mar 23 2025