Finite dimensional representations of Sl(2)

수학노트
http://bomber0.myid.net/ (토론)님의 2010년 4월 10일 (토) 12:52 판
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introduction

 

 

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

 

 

Hermite reciprocity
  • [GW1998]
  • symme

 

symmetric power of sl(2) representations
  • borrowed from sl(2) - orthogonal polynomials and Lie theory
  • 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

 

 

related items

 

 

encyclopedia

 

 

books

 

 

articles

[[2010년 books and articles|]]

 

 

question and answers(Math Overflow)

 

 

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