"Finite dimensional representations of Sl(2)"의 두 판 사이의 차이
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specialization==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==introduction | + | ==introduction== |
* [[affine sl(2) $A^{(1)} 1$]] | * [[affine sl(2) $A^{(1)} 1$]] | ||
| 8번째 줄: | 8번째 줄: | ||
| − | <h5 style="line-height: 2em; margin: 0px;">specialization | + | <h5 style="line-height: 2em; margin: 0px;">specialization== |
* Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br> | * Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br> | ||
| 17번째 줄: | 17번째 줄: | ||
| − | <h5 style="line-height: 2em; margin: 0px;">representation theory | + | <h5 style="line-height: 2em; margin: 0px;">representation theory== |
* integrable weights and Weyl vector<br><math>\omega=\frac{1}{2}\alpha</math><br><math>\rho=\omega</math><br> | * integrable weights and Weyl vector<br><math>\omega=\frac{1}{2}\alpha</math><br><math>\rho=\omega</math><br> | ||
| 28번째 줄: | 28번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">character formula and Chebyshev polynomial of the 2nd kind | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">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> | * <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> | ||
| 38번째 줄: | 38번째 줄: | ||
| − | ==Hermite reciprocity | + | ==Hermite reciprocity== |
* '''[GW1998]''' | * '''[GW1998]''' | ||
| 47번째 줄: | 47번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">symmetric power of sl(2) representations | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">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> | * 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> | ||
| 86번째 줄: | 86번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">exterior algebra of sl(2) representations | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">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> | * 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> | ||
| 103번째 줄: | 103번째 줄: | ||
| − | ==Clebsch-Gordan coefficients | + | ==Clebsch-Gordan coefficients== |
* [[3j symbol (Clebsch-Gordan coefficient)]] | * [[3j symbol (Clebsch-Gordan coefficient)]] | ||
| 111번째 줄: | 111번째 줄: | ||
| − | ==Catalan numbers | + | ==Catalan numbers== |
* http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/ | * http://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/ | ||
| 124번째 줄: | 124번째 줄: | ||
| − | ==history | + | ==history== |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 132번째 줄: | 132번째 줄: | ||
| − | ==related items | + | ==related items== |
* [[affine sl(2) $A^{(1)} 1$]] | * [[affine sl(2) $A^{(1)} 1$]] | ||
| 144번째 줄: | 144번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">encyclopedia | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">encyclopedia== |
* [http://pythagoras0.springnote.com/pages/4783755 q-이항정리] | * [http://pythagoras0.springnote.com/pages/4783755 q-이항정리] | ||
| 156번째 줄: | 156번째 줄: | ||
| − | ==books | + | ==books== |
* '''[GW1998]'''Goodman and Wallach,Representations and invariants of the classical groups | * '''[GW1998]'''Goodman and Wallach,Representations and invariants of the classical groups | ||
| 168번째 줄: | 168번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">articles | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103);">articles== |
[[2010년 books and articles|2010년 books and articles]] | [[2010년 books and articles|2010년 books and articles]] | ||
| 186번째 줄: | 186번째 줄: | ||
| − | ==question and answers(Math Overflow) | + | ==question and answers(Math Overflow)== |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 195번째 줄: | 195번째 줄: | ||
| − | ==blogs | + | ==blogs== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 205번째 줄: | 205번째 줄: | ||
| − | ==experts on the field | + | ==experts on the field== |
* http://arxiv.org/ | * http://arxiv.org/ | ||
| 213번째 줄: | 213번째 줄: | ||
| − | ==links | + | ==links== |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
2012년 10월 28일 (일) 15:41 판
introduction
- [[affine sl(2) $A^{(1)} 1$]]
- quantum sl(2)
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 \(V_k\) with the highest integrable weight \(\lambda=k\omega\)
- Weyl-Kac 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==
- \(U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)\)
U_0[x]=1
U_1[x]=2 x
U_2[x]=-1+4 x^2
U_3[x]=-4 x+8 x^3
- character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials
\(U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}\)
- \(w=e^{i\theta}\),\(z=w+w^{-1}=2\cos\theta\)
\(p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_{2}(z)=z^2-1\)
\(p_{3}(z)=z^3-2z\)
\(p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)\)
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
- [[affine sl(2) $A^{(1)} 1$]]
- cyclotomic numbers and Chebyshev polynomials
- Weyl-Kac character formula
- Macdonald constant term conjecture
encyclopedia==
- q-이항정리
- 체비셰프 다항식
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- [GW1998]Goodman and Wallach,Representations and invariants of the classical groups
- 2010년 books and articles
- 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
- SL(2,C), SU(2), and Chebyshev polynomials
- 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/~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)
blogs
experts on the field
links
\(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\)
\(\Phi=\{\alpha,-\alpha\}\)
- integrable weights and Weyl vector
\(\omega=\frac{1}{2}\alpha\)
\(\rho=\omega\) - there is a unique k+1 dimensional irreducible module \(V_k\) with the highest integrable weight \(\lambda=k\omega\)
- Weyl-Kac formula
\(\operatorname{ch}L(k\omega)=\frac{e^{(k+1)\omega}-e^{-(k+1)\omega}}{e^{\omega}-e^{-\omega}}=e^{k\omega}+e^{(k-2)\omega}+\cdots+e^{-k\omega}\)
- \(U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)\)
U_0[x]=1
U_1[x]=2 x
U_2[x]=-1+4 x^2
U_3[x]=-4 x+8 x^3 - character evaluated at an element of SU(2) with the eigenvalues e^{i\theta}, e^{-i\theta} is given by the Chebyshev polynomials
\(U_k(\cos\theta)= \frac{\sin (k+1)\theta}{\sin \theta}\) - \(w=e^{i\theta}\),\(z=w+w^{-1}=2\cos\theta\)
\(p_k(z)=\frac{w^{k+1}-w^{-k-1}}{w-w^{-1}}\)
\(p_{0}(z)=1\)
\(p_{1}(z)=z\)
\(p_{2}(z)=z^2-1\)
\(p_{3}(z)=z^3-2z\)
\(p_k(z)^2=1+p_{k-1}(z)p_{k+1}(z)\)
- 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
\([n]_{q}=\frac{q^n-q^{-n}}{q-q^{-1}}\)
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. ■
\(\prod_{j=0}^{k}(1+zq^{k-2j})}=\sum_{j=0}^{k+1}\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}z^j\)
\(\begin{bmatrix} k+1 \\ j\end{bmatrix}_{q}q^{j(j-1)/2}\)
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
- [[affine sl(2) $A^{(1)} 1$]]
- cyclotomic numbers and Chebyshev polynomials
- Weyl-Kac character formula
- Macdonald constant term conjecture
encyclopedia==
- q-이항정리
- 체비셰프 다항식
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- [GW1998]Goodman and Wallach,Representations and invariants of the classical groups
- 2010년 books and articles
- 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
- SL(2,C), SU(2), and Chebyshev polynomials
- 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/~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)
blogs
experts on the field
links
- SL(2,C), SU(2), and Chebyshev polynomials
- 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/~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)
blogs
experts on the field