Finite dimensional representations of Sl(2)

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

 

 

specialization

• Cartan matrix
  \(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\)
  
• root system
  \(\Phi=\{\alpha,-\alpha\}\)
  
• 
  
  

 

 

representation theory

 

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

• there is a unique k+1 dimensional irreducible module \(V_k\)
  

 

 

 

character formula and Chebyshev polynomial of the 2nd kind

• 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
  \({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)
  

• \(U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)\)
  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
  \(U_n(\cos\theta)= \frac{\sin (n+1)\theta}{\sin \theta}\)
  
• \(w=e^{i\theta}\), \(z=w+w^{-1}=2\cos\theta\)
  \(p_i(z)=\frac{w^{i+1}-w^{-i-1}}{w-w^{-1}}\)
  \(p_{0}(z)=1\)
  \(p_{1}(z)=z\)
  \(p_{2}(z)=z^2-1\)
  \(p_{3}(z)=z^3-2z\)
  \(p_i(z)^2=1+p_{i-1}(z)p_{i+1}(z)\)
  

 

 

Hermite reciprocity

• [GW1998]
• dimension of symmetric algebra and exterior algebra of V_k

 

 

symmetric power of sl(2) representations

• q-binomial type formula
  \(\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}=\sum_{j=0}^{\infty}t^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}\)
  
•  
  the character of j-th symmetric power of V_k is
  \(\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}\)
  where the q-analogue of the natural number is defined as 
  \([n]_{q}=\frac{q^n-q^{-n}}{q-q^{-1}}\)
  

 

(proof)

Fix a k throughout the argument.

Let \(F_j(q)\) be the character of j-th symmetric power of V_k.

\(F_j(q)=\sum_{m_0,\cdots,m_k}q^{(k-0)m_0+(k-2)m_1+\cdots+(2-k)m_{k-1}+(0-k)m_k}\)

where \(m_0+m_1+\cdots+m_k=j\)

Now consider the generating function

\(F(t,q)=\sum_{j=0}^{\infty}F_j(q)t^j\)

I claim that

\(F(t,q)=\sum_{j=0}^{\infty}F_j(q)t^j=\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}\). 

To prove that see the power series expansion of a factor\[(1-tq^{k-2j})^{-1}=\sum_{m=0}^{\infty}t^mq^{m(k-2j)}\]. Therefore

\(\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}=\sum_{m_0,\cdots,m_k}t^{m_0+\cdots+m_k}q^{(k-0)m_0+(k-2)m_1+\cdots+(2-k)m_{k-1}+(0-k)m_k}\)

Now we can easily check

\(\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}=\sum_{j=0}^{\infty}t^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}\)

■

 

 

 

exterior algebra of sl(2) representations

• q-binomial type formula
  \(\prod_{j=0}^{k}(1+tq^{k-2j})}=\sum_{j=0}^{k+1}\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}t^j\)
  
• the character of j-th exterior algebra of V_k is
  \(\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}\)
  

■

 

 

Catalan numbers

1. 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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• cyclotomic numbers and Chebyshev polynomials
• Weyl-Kac character formula
• Macdonald constant term conjecture

 

 

encyclopedia

• q-이항정리
• 체비셰프 다항식
• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

• [GW1998]Representations and invariants of the classical groups
  
  • Goodman and Wallach
• 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)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표 현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/