Finite dimensional representations of Sl(2)

introduction

• affine sl(2)
• quantum sl(2)

 

 

specialization

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

 

 

representation theory

• integrable weights and Weyl vector
  \(\omega=\frac{1}{2}\alpha\)
  \(\rho=\omega\)
  
• there is a unique k+1 dimensional irreducible module \(V_k\) with the highest integrable weight \(\lambda=k\omega\)
  

• Weyl-Kac formula
  \(\operatorname{ch}L(k\omega)=\frac{e^{(k+1)\omega}-e^{-(k+1)\omega}}{e^{\omega}-e^{-\omega}}=e^{k\omega}+e^{(k-2)\omega}+\cdots+e^{-k\omega}\)
  

 

 

character formula and 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
  \(U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}\)
  
• \(w=e^{i\theta}\),\(z=w+w^{-1}=2\cos\theta\)
  \(p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}\)
  \(p_{0}(z)=1\)
  \(p_{1}(z)=z\)
  \(p_{2}(z)=z^2-1\)
  \(p_{3}(z)=z^3-2z\)
  \(p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)\)
  

 

 

Hermite reciprocity

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

 

 

symmetric power of sl(2) representations

• q-binomial type formula (Heine formula,useful techniques in q-series)
  \(\prod_{j=0}^{k}(1-zq^{k-2j})^{-1}=\sum_{j=0}^{\infty}z^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}\)
  
• 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

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

I claim that

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

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

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

Now we can easily check

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

 

 

exterior algebra of sl(2) representations

• q-binomial type formula (Gauss formula,useful techniques in q-seriesq-analogue of summation formulas)

\[\prod_{j=0}^{k}(1+zq^{k-2j})=\sum_{j=0}^{k+1}\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}z^j\]

• the character of j-th exterior algebra of V_k is \[\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}\]
  

 

(proof)

analogous to the above. ■

 

 

 

Clebsch-Gordan coefficients

• 3j symbol (Clebsch-Gordan coefficient)

 

 

Catalan numbers

• 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

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

• affine sl(2)
• Weyl-Kac character formula
• Macdonald constant term conjecture

 

 

encyclopedia

• q-이항정리
• 체비셰프 다항식

 

 

books

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

articles

• Bacry, Henri. 1987. “SL(2,C), SU(2), and Chebyshev Polynomials.” Journal of Mathematical Physics 28 (10) (October 1): 2259–2267. doi:10.1063/1.527759.