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

2020년 12월 28일 (월) 05:21 기준 최신판

introduction

Catalan numbers

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}]

@@ 3번째 줄: / 3번째 줄: @@
 * [[affine sl(2)]]
 * [[quantum sl(2)]]
+* [[Macdonald constant term conjecture]]
+* {{수학노트|url=리대수 sl(2,C)의 유한차원 표현론}}
-==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]])
-:<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
-Fix a k throughout the argument.
-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>
-where <math>m_0+m_1+\cdots+m_k=j</math>
-Now consider the generating function
-:<math>F(z,q)=\sum_{j=0}^{\infty}F_j(q)z^j</math>
-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>
-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>
-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>■
-===exterior algebra of sl(2) representations===
-* 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>
-* 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>
-;proof
-analogous to the above. ■
-==Clebsch-Gordan coefficients==
-* [[3j symbol (Clebsch-Gordan coefficient)]]
 ==Catalan numbers==
@@ 75번째 줄: / 12번째 줄: @@
 * 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}]<br>Table[Simplify[f[2 k]], {k, 1, 10}]<br>Table[CatalanNumber[n], {n, 1, 10}]
+# 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==
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
-==related items==
-* [[affine sl(2)]]
-* [[Weyl-Kac character formula]]
-* [[Macdonald constant term conjecture]]
-==encyclopedia==
-* [http://pythagoras0.springnote.com/pages/4783755 q-이항정리]
-* [http://pythagoras0.springnote.com/pages/4682477 체비셰프 다항식]
-==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.
 [[분류:개인노트]]
 [[분류:math and physics]]
 [[분류:Lie theory]]
+[[분류:migrate]]

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

2020년 12월 28일 (월) 05:21 기준 최신판

introduction

Catalan numbers

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