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

1,3

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

Example

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.

Prog

(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

Crossrefs

Sequence in context: A348326 A364213 A051907 * A093569 A384905 A393825

Adjacent sequences: A178173 A178174 A178175 * A178177 A178178 A178179

Keyword

nonn

Author

Andrew Rupinski, Dec 18 2010

Extensions

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

Status

approved