"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이
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2013년 12월 14일 (토) 16:04 판
introduction
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
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
- 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
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.