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

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imported>Pythagoras0
잔글 (Pythagoras0 사용자가 Sl(2) - orthogonal polynomials and Lie theory 문서를 Finite dimensional representations of Sl(2) 문서로 옮겼습니다.)
imported>Pythagoras0
6번째 줄: 6번째 줄:
 
 
 
 
  
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==Hermite reciprocity==
 
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* '''[GW1998]'''
 
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* dimension of symmetric algebra and exterior algebra of $V_k$
 
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===symmetric power of sl(2) representations===
 
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* q-binomial type formula (Heine formula,[[useful techniques in q-series]])
 
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:<math>\prod_{j=0}^{k}(1-zq^{k-2j})^{-1}=\sum_{j=0}^{\infty}z^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math><br>
 
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* the character of j-th symmetric power of $V_k$ is
 
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:<math>\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math>
 
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where the q-analogue of the natural number is defined as <math>[n]_{q}=\frac{q^n-q^{-n}}{q-q^{-1}}</math>
 
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;proof
 
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Fix a k throughout the argument.
 
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Let <math>F_j(q)</math> be the character of j-th symmetric power of $V_k$.
 
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:<math>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}</math>
 
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where <math>m_0+m_1+\cdots+m_k=j</math>
 
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Now consider the generating function
 
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:<math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j</math>
 
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I claim that
 
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:<math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j=\prod_{j=0}^{k}(1-zq^{k-2j})^{-1}</math>
 
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To prove that see the power series expansion of a factor
 
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:<math>(1-zq^{k-2j})^{-1}=\sum_{m=0}^{\infty}z^mq^{m(k-2j)}</math>
 
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Therefore
 
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:<math>\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}</math>
 
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Now we can easily check
 
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:<math>\prod_{j=0}^{k}(1-zq^{k-2j})^{-1}=\sum_{j=0}^{\infty}z^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math>■
 
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===exterior algebra of sl(2) representations===
 
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* q-binomial type formula (Gauss formula,[[useful techniques in q-series]][[q-analogue of summation formulas|q-analogue of summation formulas]])
 
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:<math>\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</math>
 
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* the character of j-th exterior algebra of $V_k$ is
 
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:<math>\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}</math>
 
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;proof
 
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analogous to the above. ■
 
  
 
==Clebsch-Gordan coefficients==
 
==Clebsch-Gordan coefficients==

2013년 12월 14일 (토) 16:08 판

introduction

 


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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