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

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

integrable weights and Weyl vector
\(\omega=\frac{1}{2}\alpha\)
\(\rho=\omega\)
integrable weights \(\lambda=n\omega\)
Weyl-Kac formula
\(\operatorname{ch}L(n\omega)=\frac{e^{(n+1)\omega}-e^{-(n+1)\omega}}{e^{\omega}-e^{-\omega}}=e^{n\omega}+e^{(n-2)\omega}+\cdots+e^{-n\omega}\)

\(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
U_0[x]=1
U_1[x]=2 x
U_2[x]=-1+4 x^2
U_3[x]=-4 x+8 x^3
character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials
\(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\)
\(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
\(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_{2}(z)=z^2-1\)
\(p_{3}(z)=z^3-2z\)
\(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)

character of j-th symmetric power of V_k is
\(\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}\)

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|]]

@@ 26번째 줄: / 26번째 줄: @@
 *  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>\operatorname{ch}L(n\omega)=\frac{e^{(n+1)\omega}-e^{-(n+1)\omega}}{e^{\omega}-e^{-\omega}}=e^{n\omega}+e^{(n-2)\omega}+\cdots+e^{-n\omega}</math><br>
+*   <br>
@@ 42번째 줄: / 43번째 줄: @@
 <h5>Hermite reciprocity</h5>
+* '''[GW1998]'''
 *  character of j-th symmetric power of V_k is<br><math>\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math><br>
@@ 90번째 줄: / 93번째 줄: @@
 <h5>books</h5>
-* Representations and invariants of the classic
+* '''[GW1998]'''Representations and invariants of the classical groups<br>
-* Goodman and Wallach
+** Goodman and Wallach
 * [[2010년 books and articles]]<br>
 * http://gigapedia.info/1/