"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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* '''[GW1998]''' | * '''[GW1998]''' | ||
| − | * dimension of symmetric algebra and exterior algebra of V_k | + | * dimension of symmetric algebra and exterior algebra of $V_k$ |
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===symmetric power of sl(2) representations=== | ===symmetric power of sl(2) representations=== | ||
| − | * q-binomial type formula (Heine formula,[[useful techniques in q-series]]) | + | * q-binomial type formula (Heine formula,[[useful techniques in q-series]]) |
| − | * the character of j-th symmetric power of V_k is | + | :<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> |
| − | + | * the character of j-th symmetric power of $V_k$ is | |
| + | :<math>\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math> | ||
| + | where the q-analogue of the natural number is defined as <math>[n]_{q}=\frac{q^n-q^{-n}}{q-q^{-1}}</math> | ||
;proof | ;proof | ||
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Fix a k throughout the argument. | Fix a k throughout the argument. | ||
| − | Let <math>F_j(q)</math> be the character of j-th symmetric power of V_k. | + | Let <math>F_j(q)</math> be the character of j-th symmetric power of $V_k$. |
| − | + | :<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> | |
| − | <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> | ||
where <math>m_0+m_1+\cdots+m_k=j</math> | where <math>m_0+m_1+\cdots+m_k=j</math> | ||
Now consider the generating function | Now consider the generating function | ||
| − | + | :<math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j</math> | |
| − | <math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j</math> | ||
I claim that | I claim that | ||
| + | :<math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j=\prod_{j=0}^{k}(1-zq^{k-2j})^{-1}</math> | ||
| − | + | To prove that see the power series expansion of a factor | |
| − | + | :<math>(1-zq^{k-2j})^{-1}=\sum_{m=0}^{\infty}z^mq^{m(k-2j)}</math> | |
| − | To prove that see the power series expansion of a factor: | + | Therefore |
| − | + | :<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> | |
| − | <math>(1-zq^{k-2j})^{-1}=\sum_{m=0}^{\infty}z^mq^{m(k-2j)}</math> | ||
| − | |||
| − | <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> | ||
Now we can easily check | Now we can easily check | ||
| − | + | :<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>■ | |
| − | <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>■ | ||
| 88번째 줄: | 85번째 줄: | ||
* q-binomial type formula (Gauss formula,[[useful techniques in q-series]][[q-analogue of summation formulas|q-analogue of summation formulas]]) | * q-binomial type formula (Gauss formula,[[useful techniques in q-series]][[q-analogue of summation formulas|q-analogue of summation formulas]]) | ||
| − | :<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 | + | :<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> |
| − | * the character of j-th exterior algebra of V_k is :<math>\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}</math | + | * the character of j-th exterior algebra of $V_k$ is |
| + | :<math>\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}</math> | ||
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analogous to the above. ■ | analogous to the above. ■ | ||
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| − | |||
==Clebsch-Gordan coefficients== | ==Clebsch-Gordan coefficients== | ||
2013년 12월 14일 (토) 14:27 판
introduction
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 \(V_k\) of highest weight \(\lambda=k\omega\)
- Weyl-Kac character 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
- \(U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)\)
- character evaluated at an element of SU(2) with the eigenvalues $e^{i\theta}, e^{-i\theta}$ is given by the Chebyshev polynomials
\[U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}\]
- \(w=e^{i\theta}\),\(z=w+w^{-1}=2\cos\theta\)
- \(p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}\)
- \(p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)\)
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
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.