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

2020년 12월 28일 (월) 04:21 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) 04:21 기준 최신판

introduction

Catalan numbers

둘러보기 메뉴

검색

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* [[affine sl(2) $A^{(1)} 1$]]
+* [[affine sl(2)]]
-* [[search?q=quantum%20sl%282%29&parent id=5522041|quantum sl(2)]]
+* [[quantum sl(2)]]
+* [[Macdonald constant term conjecture]]
+* {{수학노트|url=리대수 sl(2,C)의 유한차원 표현론}}
-==specialization==
-* Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br>
-* root system<br><math>\Phi=\{\alpha,-\alpha\}</math><br>
-==representation theory==
-* 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 character formula|Weyl-Kac formula]]<br><math>\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}</math><br>
-==character formula and Chebyshev polynomial of the 2nd kind==
-* <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>
-==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]])<br><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<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(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]])<br><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><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>
-(proof)
-analogous to the above. ■
-==Clebsch-Gordan coefficients==
-* [[3j symbol (Clebsch-Gordan coefficient)]]
 ==Catalan numbers==
@@ 116번째 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) $A^{(1)} 1$]]
-* [[cyclotomic numbers and Chebyshev polynomials]]
-* [[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]
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
-==books==
-* '''[GW1998]'''Goodman and Wallach,Representations and invariants of the classical groups
-* [[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=
-==articles==
-[[2010년 books and articles|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
-==question and answers(Math Overflow)==
-* http://mathoverflow.net/search?q=
-* http://mathoverflow.net/search?q=
-==blogs==
-* 구글 블로그 검색<br>
-** http://blogsearch.google.com/blogsearch?q=
-** http://blogsearch.google.com/blogsearch?q=
-==experts on the field==
-* http://arxiv.org/
-==links==
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표 현 안내]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* http://functions.wolfram.com/
 [[분류:개인노트]]
 [[분류:math and physics]]
+[[분류:Lie theory]]
+[[분류:migrate]]