"Elements of finite order (EFO) in Lie groups"의 두 판 사이의 차이

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==introduction==
 
==introduction==
 
* explicit formulas for the number of conjugacy classes of EFOs in Lie groups
 
* explicit formulas for the number of conjugacy classes of EFOs in Lie groups
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* appears for the number of certain vacua in the quantum moduli space of M-theory compactifications on manifolds of $G_2$ holonomy
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* appears for the number of certain vacua in the quantum moduli space of M-theory compactifications on manifolds of <math>G_2</math> holonomy
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* $N(G,m)$ : number of conjugacy classes of $G$ in $E(G,m)$
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* <math>N(G,m)</math> : number of conjugacy classes of <math>G</math> in <math>E(G,m)</math>
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* $N(G,m,s)$ : number of conjugacy classes of $G$ in $E(G,m,s)$
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* <math>N(G,m,s)</math> : number of conjugacy classes of <math>G</math> in <math>E(G,m,s)</math>
  
 
==EFO in unitary groups==
 
==EFO in unitary groups==
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===$U(n)$===
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===<math>U(n)</math>===
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* $N(G,m)= {n+m-1\choose m-1}$
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* <math>N(G,m)= {n+m-1\choose m-1}</math>
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* $N(G,m,s)=\frac{s}{n}{n\choose s}{m\choose s}$
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* <math>N(G,m,s)=\frac{s}{n}{n\choose s}{m\choose s}</math>
  
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===$SU(n)$===
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===<math>SU(n)</math>===
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* $N(G,m)= \frac{1}{m}{n+m-1\choose m-1}$ if $(n,m)=1$
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* <math>N(G,m)= \frac{1}{m}{n+m-1\choose m-1}</math> if <math>(n,m)=1</math>
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* $N(G,m,s)= \frac{s}{nm}{n\choose s}{m\choose s}$ if $(n,m)=1$
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* <math>N(G,m,s)= \frac{s}{nm}{n\choose s}{m\choose s}</math> if <math>(n,m)=1</math>
  
  
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==computational resource==
 
==computational resource==
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxLU5vUzJRQUNGdnc/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxLU5vUzJRQUNGdnc/edit
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===OEIS===
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* type A http://oeis.org/A008610
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* type C http://oeis.org/A005993
  
  
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2020년 11월 14일 (토) 01:03 기준 최신판

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


computational resource

OEIS


questions

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