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

2010년 4월 5일 (월) 06:10 판

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
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)

\(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
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 4월 5일 (월) 05:36 판 (원본 보기) http://bomber0.myid.net/ (토론) ← 이전 편집		2010년 4월 5일 (월) 06:10 판 (원본 보기) http://bomber0.myid.net/ (토론) 다음 편집 →
23번째 줄:		23번째 줄:
	* character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials<br><math>U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}</math><br>		* character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials<br><math>U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}</math><br>
	* <math>w=e^{i\theta}</math>, <math>z=w+w^{-1}=2\cos\theta</math><br><math>p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}</math><br><math>p_{0}(z)=1</math><br><math>p_{1}(z)=z</math><br><math>p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)</math><br>		* <math>w=e^{i\theta}</math>, <math>z=w+w^{-1}=2\cos\theta</math><br><math>p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}</math><br><math>p_{0}(z)=1</math><br><math>p_{1}(z)=z</math><br><math>p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)</math><br>
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