"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이

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<h5>introduction</h5>
 
<h5>introduction</h5>
  
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* http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/
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* [http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/ ]http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/
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* http://mathoverflow.net/questions/17197/how-does-this-relationship-between-the-catalan-numbers-and-su2-generalize
  
 
 
 
 

2010년 4월 5일 (월) 05:36 판

introduction

 

 

character formula of sl(2)
  • Weyl-Kac formula
    \(ch(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}\)
  • for trivial representation, we get denominator identity
    \({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)

 

 

Chebyshev polynomial of the 2nd kind
  • \(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
  • character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials
    \(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\)
  • \(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
    \(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
    \(p_{0}(z)=1\)
    \(p_{1}(z)=z\)
    \(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)

 

 

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[[2010년 books and articles|]]

 

 

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