Finite dimensional representations of Sl(2)
http://bomber0.myid.net/ (토론)님의 2010년 4월 5일 (월) 13:58 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 of sl(2)
- 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
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)
Chebyshev polynomial of the 2nd kind
- \(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
- character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials
\(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\) - \(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
\(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)
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
- 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