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

questions

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