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

2010년 4월 5일 (월) 04:17 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
\(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\)
\(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
\(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)

[[2010년 books and articles|]]

@@ 20번째 줄: / 20번째 줄: @@
 * <math>U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)</math>
-*  character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is<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>,<br><math>p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}</math> for <math> i=1,\cdots, k</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>p_{0}(z)=1</math>
-* <math>p_{1}(z)=z</math>
-* <math>p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)</math>
@@ 43번째 줄: / 39번째 줄: @@
 * [[cyclotomic numbers and Chebyshev polynomials]]
 * [[Weyl-Kac character formula]]
+* [[Macdonald constant term conjecture]]