Elements of finite order (EFO) in Lie groups

introduction

• explicit formulas for the number of conjugacy classes of EFOs in Lie groups
• appears for the number of certain vacua in the quantum moduli space of M-theory compactifications on manifolds of \(G_2\) holonomy
• \(N(G,m)\) : number of conjugacy classes of \(G\) in \(E(G,m)\)
• \(N(G,m,s)\) : number of conjugacy classes of \(G\) in \(E(G,m,s)\)

EFO in unitary groups

\(U(n)\)

• \(N(G,m)= {n+m-1\choose m-1}\)
• \(N(G,m,s)=\frac{s}{n}{n\choose s}{m\choose s}\)

\(SU(n)\)

• \(N(G,m)= \frac{1}{m}{n+m-1\choose m-1}\) if \((n,m)=1\)
• \(N(G,m,s)= \frac{s}{nm}{n\choose s}{m\choose s}\) if \((n,m)=1\)

related items

• simple Lie groups and the Legendre symbol
• 틀:수학노트

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxLU5vUzJRQUNGdnc/edit

OEIS

• type A http://oeis.org/A008610
• type C http://oeis.org/A005993

questions

• http://mathoverflow.net/questions/86499/representations-of-quantum-groups-at-roots-of-unity