"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이

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imported>Pythagoras0
imported>Pythagoras0
3번째 줄: 3번째 줄:
 
* [[affine sl(2)]]
 
* [[affine sl(2)]]
 
* [[quantum sl(2)]]
 
* [[quantum sl(2)]]
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==specialization==
 
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* Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br>
 
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* root system<br><math>\Phi=\{\alpha,-\alpha\}</math><br>
 
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==representation theory==
 
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*  integrable weights and Weyl vector
 
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:<math>\omega=\frac{1}{2}\alpha, \rho=\omega</math>
 
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*  there is a unique k+1 dimensional irreducible module <math>V_k</math> of highest weight <math>\lambda=k\omega</math>
 
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* [[Weyl-Kac character formula]]
 
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:<math>\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}</math>
 
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===character formula and Chebyshev polynomial of the 2nd kind===
 
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* <math>U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)</math>
 
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* character evaluated at an element of SU(2) with the eigenvalues $e^{i\theta}, e^{-i\theta}$ is given by the Chebyshev polynomials
 
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:<math>U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}</math>
 
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* <math>w=e^{i\theta}</math>,<math>z=w+w^{-1}=2\cos\theta</math>
 
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* <math>p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}</math>
 
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* <math>p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)</math><br>
 
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2013년 12월 14일 (토) 14:48 판

introduction

 

Hermite reciprocity

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

 

 

symmetric power of sl(2) representations

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

  • 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

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

  • 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

 

 

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

  • [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.

 

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