"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이
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| 25번째 줄: | 25번째 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> | * 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}}</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> |
| 35번째 줄: | 35번째 줄: | ||
<h5 style="line-height: 2em; margin: 0px;">Chebyshev polynomial of the 2nd kind</h5> | <h5 style="line-height: 2em; margin: 0px;">Chebyshev polynomial of the 2nd kind</h5> | ||
| − | * <math>U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)</math> | + | * <math>U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)</math><br> U_0[x]=1<br> U_1[x]=2 x<br> U_2[x]=-1+4 x^2<br> U_3[x]=-4 x+8 x^3<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> | * 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_{2}(z)=z^2-1</math><br><math>p_{ | + | * <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_{2}(z)=z^2-1</math><br><math>p_{3}(z)=z^3-2z</math><br><math>p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)</math><br> |
| 73번째 줄: | 73번째 줄: | ||
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">encyclopedia</h5> | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">encyclopedia</h5> | ||
| + | * [http://pythagoras0.springnote.com/pages/4682477 체비셰프 다항식] | ||
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
* http://www.scholarpedia.org/ | * http://www.scholarpedia.org/ | ||
2010년 4월 9일 (금) 06:36 판
introduction
- http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/
- http://mathoverflow.net/questions/17197/how-does-this-relationship-between-the-catalan-numbers-and-su2-generalize
character formula
- 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
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)
specialization
- Cartan matrix
\(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\) - root system
\(\Phi=\{\alpha,-\alpha\}\)
- integrable weights and Weyl vector
\(\omega=\frac{1}{2}\alpha\)
\(\rho=\omega\)
integrable weights \(\lambda=n\omega\) - Weyl-Kac formula
\(\operatorname{ch}L(n\omega)=\frac{e^{(n+1)\omega}-e^{-(n+1)\omega}}{e^{\omega}-e^{-\omega}}=e^{n\omega}+e^{(n-2)\omega}+\cdots+e^{-n\omega}\)
Chebyshev polynomial of the 2nd kind
- \(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
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
\(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\) - \(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
\(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_{2}(z)=z^2-1\)
\(p_{3}(z)=z^3-2z\)
\(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)
Catalan numbers
- 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}]
history
- cyclotomic numbers and Chebyshev polynomials
- Weyl-Kac character formula
- Macdonald constant term conjecture
encyclopedia
- 체비셰프 다항식
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
[[2010년 books and articles|]]
- SL(2,C), SU(2), and Chebyshev polynomials
- Henri Bacry, J. Math. Phys. 28, 2259 (1987)
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- http://dx.doi.org/10.1063/1.527759
question and answers(Math Overflow)
blogs
experts on the field