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

2010년 4월 9일 (금) 05:48 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
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)

integrable weights and Weyl vector
\(\omega=\frac{1}{2}\alpha\)
\(\rho=\omega\)
integrable weights \(\lambda=n\omega\)
Weyl-Kac formula
\(ch(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}\)

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

f[n_] := Integrate[(2 Cos[Pi*x])^n*2 (Sin[Pi*x])^2, {x, 0, 1}]
Table[Simplify[f[2 k]], {k, 1, 10}]
Table[CatalanNumber[n], {n, 1, 10}]

[[2010년 books and articles|]]

@@ 8번째 줄: / 8번째 줄: @@
-<h5>character formula of sl(2)</h5>
+<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>
+*  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>
-*  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;">specialization</h5>
-* [[affine sl(2) $A^{(1)} 1$|affine sl(2)]]<br>
+*  Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br>
+*  Find the highest root <br><math>\alpha</math><br>
+*  integrable weights and Weyl vector<br><math>\omega=\frac{1}{2}\alpha</math><br><math>\rho=\omega</math><br> integrable weights <math>\lambda=n\omega</math><br>
+*  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>