"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}]

"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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-<h5>introduction</h5>
+==introduction==
-* http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/
+* [[affine sl(2)]]
-* http://mathoverflow.net/questions/17197/how-does-this-relationship-between-the-catalan-numbers-and-su2-generalize
+* [[quantum sl(2)]]
-* [[affine sl(2) $A^{(1)} 1$]]
-* [[search?q=quantum%20sl(2)&parent id=5522041|quantum sl(2)]]
-<h5 style="line-height: 2em; margin: 0px;">specialization</h5>
-*  Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br>
-*  root system<br><math>\Phi=\{\alpha,-\alpha\}</math><br>
-<h5 style="line-height: 2em; margin: 0px;">representation theory</h5>
-*  integrable weights and Weyl vector<br><math>\omega=\frac{1}{2}\alpha</math><br><math>\rho=\omega</math><br>
-*  there is a unique k+1 dimensional irreducible module <math>V_k</math> with the highest integrable weight <math>\lambda=k\omega</math><br>
-*  Weyl-Kac formula<br><math>\operatorkame{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><br>
-<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">character formula and Chebyshev polynomial of the 2nd kind</h5>
-* <math>U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x)</math><br> U_0[x]=1<br> U_1[x]=2 x<br> U_2[x]=-1+4 x^2<br> U_3[x]=-4 x+8 x^3<br>
-*  character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials<br><math>U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}</math><br>
-* <math>w=e^{i\theta}</math>, <math>z=w+w^{-1}=2\cos\theta</math><br><math>p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}</math><br><math>p_{0}(z)=1</math><br><math>p_{1}(z)=z</math><br><math>p_{2}(z)=z^2-1</math><br><math>p_{3}(z)=z^3-2z</math><br><math>p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)</math><br>
-<h5>Hermite reciprocity</h5>
-* '''[GW1998]'''
-* dimension of symmetric algebra and exterior algebra of V_k
-<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">symmetric power of sl(2) representations</h5>
-*  q-binomial type formula<br><math>\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}=\sum_{j=0}^{\infty}t^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math><br>
-*  the character of j-th symmetric power of V_k is<br><math>\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math><br> where the q-analogue of the natural number is defined as <br><math>[n]_{q}=\frac{q^n-q^{-n}}{q-q^{-1}}</math><br>
-(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(t,q)=\sum_{j=0}^{\infty}F_j(q)t^j</math>
-I claim that
-<math>F(t,q)=\sum_{j=0}^{\infty}F_j(q)t^j=\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}</math>.
-To prove that see the power series expansion of a factor:
-<math>(1-tq^{k-2j})^{-1}=\sum_{m=0}^{\infty}t^mq^{m(k-2j)}</math>. Therefore
-<math>\prod_{j=0}^{k}(1-tq^{k-2j})^{-1}=\sum_{m_0,\cdots,m_k}t^{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-tq^{k-2j})^{-1}=\sum_{j=0}^{\infty}t^j\begin{bmatrix} k+j\\ k\end{bmatrix}_{q}</math>■
-<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">exterior algebra of sl(2) representations</h5>
-*  q-binomial type formula (Gauss formula)<br><math>\prod_{j=0}^{k}(1+tq^{k-2j})}=\sum_{j=0}^{k+1}\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}t^j</math><br>
-*  the character of j-th exterior algebra of V_k is<br><math>\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}</math><br>
-<h5>Clebsch-Gordan coefficients</h5>
-<h5>Catalan numbers</h5>
-# 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}]
-<h5>history</h5>
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
-<h5>related items</h5>
-* [[cyclotomic numbers and Chebyshev polynomials]]
-* [[Weyl-Kac character formula]]
 * [[Macdonald constant term conjecture]]
+* {{수학노트|url=리대수 sl(2,C)의 유한차원 표현론}}
+==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
-<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">encyclopedia</h5>
-* [http://pythagoras0.springnote.com/pages/4783755 q-이항정리]
-* [http://pythagoras0.springnote.com/pages/4682477 체비셰프 다항식]
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
-<h5>books</h5>
-* '''[GW1998]'''Representations and invariants of the classical groups<br>
-** Goodman and Wallach
-* [[2010년 books and articles]]<br>
-* http://gigapedia.info/1/
-* http://gigapedia.info/1/
-* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
-<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">articles</h5>
-[[2010년 books and articles|]]
-* [http://dx.doi.org/10.1063/1.527759 SL(2,C), SU(2), and Chebyshev polynomials]<br>
-** Henri Bacry, J. Math. Phys. 28, 2259 (1987)
-* http://www.ams.org/mathscinet
-* http://www.zentralblatt-math.org/zmath/en/
-* http://pythagoras0.springnote.com/
-* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
-* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
-* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
-* http://dx.doi.org/10.1063/1.527759
-<h5>question and answers(Math Overflow)</h5>
-* http://mathoverflow.net/search?q=
-* http://mathoverflow.net/search?q=
-<h5>blogs</h5>
-*  구글 블로그 검색<br>
-** http://blogsearch.google.com/blogsearch?q=
-** http://blogsearch.google.com/blogsearch?q=
-<h5>experts on the field</h5>
-* http://arxiv.org/
+# 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}]
-<h5>links</h5>
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+[[분류:개인노트]]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표 현 안내]
+[[분류:math and physics]]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
+[[분류:Lie theory]]
-* http://functions.wolfram.com/
+[[분류:migrate]]