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

2010년 4월 4일 (일) 19:31 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
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)

\(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\) for \( i=1,\cdots, k\)

[[2010년 books and articles|]]

@@ 1번째 줄: / 1번째 줄: @@
 <h5>introduction</h5>
-Define <math>w^{2(2k+3)}=1</math> and <math>z=w+w^{-1}</math>
+<h5>character formula</h5>
+*  Weyl-Kac formula<br><math>ch(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}</math><br>
-<math>p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}</math> for <math> i=1,\cdots, k</math>
+*  for trivial representation, we get denominator identity<br><math>{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}</math><br>
+<h5 style="line-height: 2em; margin: 0px;">Chebyshev polynomial</h5>
 * <math>U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)</math>
+* Define <math>w^{2(2k+3)}=1</math> and <math>z=w+w^{-1}</math>
+<math>p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}</math> for <math> i=1,\cdots, k</math>
-<h5>recurrence relation</h5>
 * <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>