"Jack polynomials"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
||
| (사용자 2명의 중간 판 13개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
It has been known since the early 1970s \cite{Calogero69,Sutherland71, | It has been known since the early 1970s \cite{Calogero69,Sutherland71, | ||
| − | Sutherland72} that \eqref{Cp} with | + | Sutherland72} that \eqref{Cp} with <math>\beta=2\gamma</math> --- |
| − | to be denoted | + | to be denoted <math>\exp(-2\gamma W)</math> in analogy with |
\eqref{Fza} --- is the absolute value squared of the ground-state | \eqref{Fza} --- is the absolute value squared of the ground-state | ||
wave function for the Schr\"odinger operator \label{CSpage} | wave function for the Schr\"odinger operator \label{CSpage} | ||
| 11번째 줄: | 11번째 줄: | ||
\] | \] | ||
This operator, known as the Calogero--Sutherland Hamiltonian, | This operator, known as the Calogero--Sutherland Hamiltonian, | ||
| − | describes a system of | + | describes a system of <math>n</math> identical quantum particles on the unit circle, |
| − | with | + | with <math>\theta_i\in[0,2\pi)</math> for <math>1\leq i\leq n</math> the (angular) |
positions of the particles. The interaction between the particles | positions of the particles. The interaction between the particles | ||
| − | is described by a | + | is described by a <math>1/r^2</math> two-body potential, |
| − | + | <math>2|\sin((\theta_i - \theta_j)/2)|</math> | |
being the cord-length between particles located at | being the cord-length between particles located at | ||
| − | + | <math>\theta_i</math> and <math>\theta_j</math>. | |
| − | B.~Sutherland \cite{Sutherland71} showed that the eigenvalue | + | B.~Sutherland \cite{Sutherland71} showed that the eigenvalue <math>E_0</math> |
corresponding to the ground-state wave function | corresponding to the ground-state wave function | ||
| − | is given by | + | is given by <math>E_0=n(n^2-1)\gamma^2/12</math>. Subsequently he showed |
\cite{Sutherland72} that the conjugated operator | \cite{Sutherland72} that the conjugated operator | ||
\begin{equation}\label{CO} | \begin{equation}\label{CO} | ||
| − | + | e^{\gamma W} (H - E_0) e^{-\gamma W} = | |
\sum_{i=1}^n \Bigl(x_i \frac{\partial}{\partial x_i} \Bigr)^2 | \sum_{i=1}^n \Bigl(x_i \frac{\partial}{\partial x_i} \Bigr)^2 | ||
+ 2\gamma \sum_{\substack{i,j=1 \\ i\neq j}}^n | + 2\gamma \sum_{\substack{i,j=1 \\ i\neq j}}^n | ||
\frac{x_i + x_j}{x_i - x_j} \, \frac{\partial}{\partial x_i}, | \frac{x_i + x_j}{x_i - x_j} \, \frac{\partial}{\partial x_i}, | ||
\end{equation} | \end{equation} | ||
| − | where | + | where <math>x_j := \exp(i \theta_j)</math>, |
admits a complete set of symmetric polynomial eigenfunctions | admits a complete set of symmetric polynomial eigenfunctions | ||
| − | + | <math>P_{\lambda}^{(1/\gamma)}(x)</math>. These polynomials, | |
| − | now referred to as Jack polynomials, depend on | + | now referred to as Jack polynomials, depend on <math>x=(x_1,\dots,x_n)</math> |
| − | and are indexed by partitions | + | and are indexed by partitions <math>\lambda</math> of at most <math>n</math> parts; |
| − | + | <math>\lambda=(\lambda_1,\dots,\lambda_n)</math> with | |
| − | + | <math>\lambda_1\geq \lambda_2\geq \dots\geq \lambda_n\geq 0</math>. | |
| − | With | + | With <math>m_{\lambda}</math> denoting the monomial symmetric |
| − | function indexed by | + | function indexed by <math>\lambda</math> and <math><</math> |
the dominance ordering on partitions, the | the dominance ordering on partitions, the | ||
Jack polynomials have the structure | Jack polynomials have the structure | ||
\begin{equation}\label{stu} | \begin{equation}\label{stu} | ||
| − | P_{\ | + | P_{\lambda}^{(1/\gamma)}(x) = m_{\lambda}(x) + |
| − | \sum_{\mu < \ | + | \sum_{\mu < \lambda} a_{\lambda\mu}\, m_{\mu}(x) |
\end{equation} | \end{equation} | ||
| − | for some coefficients | + | for some coefficients <math>a_{\lambda\mu}=a_{\lambda\mu}(\gamma)</math>. |
One fundamental property of the Jack polynomials is that they | One fundamental property of the Jack polynomials is that they | ||
| 51번째 줄: | 51번째 줄: | ||
\langle f,g \rangle_{\gamma}:=\frac{1}{(2\pi)^n} | \langle f,g \rangle_{\gamma}:=\frac{1}{(2\pi)^n} | ||
\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} | \int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} | ||
| − | f( | + | f(e^{i \theta}) g(e^{-i \theta}) |
\prod_{1\le i<j\le n} | \prod_{1\le i<j\le n} | ||
| − | + | |e^{i\theta_i}-e^{i\theta_j}|^{2\gamma} \, | |
| − | \ | + | \,d \theta_1 \cdots \,d \theta_n, |
\end{equation} | \end{equation} | ||
| − | where | + | where <math>f(e^{i \theta})= |
| − | f( | + | f(e^{i \theta_1},\dots,e^{i \theta_n})</math>. |
To state the orthogonality as well as the quadratic norm evaluation let | To state the orthogonality as well as the quadratic norm evaluation let | ||
\[ | \[ | ||
| − | \Poch{b}{\ | + | \Poch{b}{\lambda}{\gamma}=\prod_{i\geq 1} (b+(1-i)\gamma)_{\lambda_i} |
\] | \] | ||
| − | with | + | with <math>(b)_n=b(b+1)\cdots(b+n-1)</math> a Pochhammer symbol. Also let |
| − | + | <math>c_{\lambda}(\gamma)</math> and <math>c_{\lambda}'(\gamma)</math> be given by | |
| − | + | :<math> | |
| + | \label{ccp} | ||
\begin{align} | \begin{align} | ||
| − | c_{\ | + | c_{\lambda}(\gamma)&=\prod_{s\in\lambda}(a(s)+l(s)\gamma+\gamma), \\ |
| − | c'_{\ | + | c'_{\lambda}(\gamma)&=\prod_{s\in\lambda}(a(s)+l(s)\gamma+1), |
\end{align} | \end{align} | ||
| − | + | </math> | |
| − | where | + | where <math>a(s)</math> and <math>l(s)</math> are the arm-length and leg-length of the |
| − | square | + | square <math>s</math> in the diagram of the partition <math>\lambda</math>, and <math>|\lambda|</math> |
| − | is the total number of boxes in the diagram of | + | is the total number of boxes in the diagram of <math>\lambda</math> \cite{Macdonald95}. |
Then | Then | ||
\begin{equation}\label{OP} | \begin{equation}\label{OP} | ||
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda}^{(1/\gamma)},P_{\mu}^{(1/\gamma)}\bigr\rangle_{\gamma} |
| − | =\delta_{\ | + | =\delta_{\lambda\mu} \: |
| − | \frac{c'_{\ | + | \frac{c'_{\lambda}(\gamma)} |
| − | {\Poch{1+(n-1)\gamma}{\ | + | {\Poch{1+(n-1)\gamma}{\lambda}{\gamma}}\: |
\frac{\Gamma(1+n\gamma)}{\Gamma^n(1+\gamma)}\: | \frac{\Gamma(1+n\gamma)}{\Gamma^n(1+\gamma)}\: | ||
| − | P_{\ | + | P_{\lambda}^{(1/\gamma)}(1^n), |
\end{equation} | \end{equation} | ||
| − | where | + | where <math>\delta_{\lambda\mu}</math> is the Kronecker delta function and |
| − | + | <math>(1^n)</math> is shorthand for <math>(1,1,\dots,1)</math>. | |
The orthogonality relation is consistent with, | The orthogonality relation is consistent with, | ||
but not an immediate consequence of the operator | but not an immediate consequence of the operator | ||
| 90번째 줄: | 91번째 줄: | ||
This degeneracy can be removed by introducing the mutually commuting | This degeneracy can be removed by introducing the mutually commuting | ||
Cherednik operators | Cherednik operators | ||
| − | + | <math>\xi_i</math> for <math>1\leq i\leq n</math> \cite{Cherednik91,Dunkl89} | |
\[ | \[ | ||
\xi_i=1-i+\frac{x_i}{\gamma} \frac{\partial}{\partial x_i} + | \xi_i=1-i+\frac{x_i}{\gamma} \frac{\partial}{\partial x_i} + | ||
| 96번째 줄: | 97번째 줄: | ||
\sum_{j=i+1}^n\frac{x_j}{x_i - x_j}\,(1 - s_{ij}), | \sum_{j=i+1}^n\frac{x_j}{x_i - x_j}\,(1 - s_{ij}), | ||
\] | \] | ||
| − | where | + | where <math>s_{ij}</math> acts by permutation |
| − | + | <math>x_i</math> and <math>x_j</math> and <math>1</math> represents the identity operator. | |
| − | Any symmetric combination of the | + | Any symmetric combination of the <math>\xi_i</math>, and in particular |
| − | + | <math>\prod_{i=1}^n (1-u\xi_i)</math>, | |
has the Jack polynomials as simultaneous eigenfunctions. | has the Jack polynomials as simultaneous eigenfunctions. | ||
| 109번째 줄: | 110번째 줄: | ||
\eqref{CNb2}, but also the more general quadratic norm evaluation | \eqref{CNb2}, but also the more general quadratic norm evaluation | ||
of the Jack polynomials corresponding to \eqref{OP} | of the Jack polynomials corresponding to \eqref{OP} | ||
| − | with | + | with <math>\lambda=\mu</math> \cite{Kakei98}. |
| − | (For | + | (For <math>\lambda=0</math> this yields \eqref{CNb1}.) |
| − | With | + | With <math>\Delta(x)</math> the Vandermonde product \eqref{VanderM} and <math>Y_{\pm}:= |
| − | \gamma^{n(n-1)/2} \prod_{1\leq i<j\leq n}(\xi_i-\xi_j\mp 1) | + | \gamma^{n(n-1)/2} \prod_{1\leq i<j\leq n}(\xi_i-\xi_j\mp 1)</math>, |
the Jack shift operators are defined by | the Jack shift operators are defined by | ||
| − | + | <math>G_{+}:=\Delta^{-1} Y_{+}</math>, <math>G_{-} = Y_{-}\Delta</math>. | |
They have an adjoint type property with respect to the inner product | They have an adjoint type property with respect to the inner product | ||
\eqref{InProd}, | \eqref{InProd}, | ||
| 122번째 줄: | 123번째 줄: | ||
Also, with | Also, with | ||
\begin{equation}\label{apm} | \begin{equation}\label{apm} | ||
| − | a_{\ | + | a_{\lambda}^{\pm}(\gamma)=\prod_{1\leq i<j\leq n} |
| − | (\ | + | (\lambda_i-\lambda_j\pm 1+(j-i\mp 1)\gamma) |
\end{equation} | \end{equation} | ||
| − | and | + | and <math>\delta</math> the staircase partition <math>(n-1,n-2,\dots,1,0)</math>, |
the shift operators act on the Jack polynomials as | the shift operators act on the Jack polynomials as | ||
| − | + | :<math>\label{fr.2} | |
\begin{align} | \begin{align} | ||
| − | G_{+} P_{\ | + | G_{+} P_{\lambda+\delta}^{(1/\gamma)}&=a_{\lambda}^{+}(\gamma+1) |
| − | P_{\ | + | P_{\lambda}^{(1/(\gamma+1))}, \\[2mm] |
| − | G_{-} P_{\ | + | G_{-} P_{\lambda}^{(1/(\gamma+1))}&=a_{\lambda}^{-}(\gamma+1) |
| − | P_{\ | + | P_{\lambda+\delta}^{(1/\gamma )}. |
\end{align} | \end{align} | ||
| − | + | </math> | |
It follows from \eqref{fr.1} and \eqref{fr.2} that | It follows from \eqref{fr.1} and \eqref{fr.2} that | ||
\[ | \[ | ||
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda}^{(1/(\gamma+1))},P_{\lambda}^{(1/(\gamma+1))} |
\bigr\rangle_{\gamma+1}= | \bigr\rangle_{\gamma+1}= | ||
| − | \frac{a_{\ | + | \frac{a_{\lambda}^{-}(\gamma+1)}{a_{\lambda}^{+}(\gamma+1)}\: |
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda+\delta}^{(1/\gamma)},P_{\lambda+\delta}^{(1/\gamma)} |
\bigr\rangle_{\gamma} | \bigr\rangle_{\gamma} | ||
\] | \] | ||
and thus | and thus | ||
\[ | \[ | ||
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda}^{(1/(\gamma+k))},P_{\lambda}^{(1/(\gamma+k))} |
\bigr\rangle_{\gamma+k}= | \bigr\rangle_{\gamma+k}= | ||
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda+k\delta}^{(1/\gamma)},P_{\lambda+k\delta}^{(1/\gamma)} |
\bigr\rangle_{\gamma}\: | \bigr\rangle_{\gamma}\: | ||
\prod_{j=1}^{k-1} | \prod_{j=1}^{k-1} | ||
| − | \frac{a_{\ | + | \frac{a_{\lambda+j\delta}^{-}(\gamma+k-j)}{a_{\lambda+j\delta}^{+}(\gamma+k-j)}. |
\] | \] | ||
| − | Taking | + | Taking <math>\gamma=0</math>, using that <math>P_{\lambda}^{(\infty)}=m_{\lambda}</math> |
(the monomial symmetric function) and | (the monomial symmetric function) and | ||
\[ | \[ | ||
| 159번째 줄: | 160번째 줄: | ||
m_{\mu}(1^n) | m_{\mu}(1^n) | ||
\] | \] | ||
| − | which is | + | which is <math>n!</math> for <math>\mu=\lambda+k\delta</math>, |
| − | it follows that for nonnegative integer | + | it follows that for nonnegative integer <math>k</math> |
\label{pageJack2} | \label{pageJack2} | ||
\begin{equation}\label{fr.3} | \begin{equation}\label{fr.3} | ||
| − | \bigl\langle P_{\ | + | \bigl\langle P_{\lambda}^{(1/k)},P_{\lambda}^{(1/k)}\bigr\rangle_k |
=n!\prod_{j=0}^{k-1} \frac{a_{\lambda+jk}^{-}(k-j)} | =n!\prod_{j=0}^{k-1} \frac{a_{\lambda+jk}^{-}(k-j)} | ||
{a_{\lambda+jk}^{+}(k-j)}. | {a_{\lambda+jk}^{+}(k-j)}. | ||
| 169번째 줄: | 170번째 줄: | ||
Using the evaluation formula \cite{Stanley89} | Using the evaluation formula \cite{Stanley89} | ||
\begin{equation}\label{ef} | \begin{equation}\label{ef} | ||
| − | P_{\ | + | P_{\lambda}^{(1/\gamma)}(1^n)=\frac{[n\gamma]_{\lambda}^{(\gamma)}} |
| − | {c_{\ | + | {c_{\lambda}(\gamma)} |
\end{equation} | \end{equation} | ||
and the definitions \eqref{ccp} and \eqref{apm} it is now | and the definitions \eqref{ccp} and \eqref{apm} it is now | ||
a straightforward exercise to verify that for | a straightforward exercise to verify that for | ||
| − | + | <math>\gamma=k</math> \eqref{OP} coincides with \eqref{fr.3}. | |
Analytic continuation off the integers is then required to | Analytic continuation off the integers is then required to | ||
establish | establish | ||
| − | \eqref{OP} for all | + | \eqref{OP} for all <math>\Re(\gamma)>-1/n</math>. |
\medskip | \medskip | ||
| 184번째 줄: | 185번째 줄: | ||
R.P.~Stanley's Cauchy identity \cite{Stanley89} | R.P.~Stanley's Cauchy identity \cite{Stanley89} | ||
\begin{equation}\label{CaP} | \begin{equation}\label{CaP} | ||
| − | \sum_{\ | + | \sum_{\lambda} \frac{c_{\lambda}(\gamma)}{c'_{\lambda}(\gamma)} \, |
| − | P_{\ | + | P_{\lambda}^{(1/\gamma)}(x) P_{\lambda}^{(1/\gamma)}(y)= |
\prod_{i=1}^n\prod_{j=1}^m (1-x_i y_j)^{-\gamma}, | \prod_{i=1}^n\prod_{j=1}^m (1-x_i y_j)^{-\gamma}, | ||
\end{equation} | \end{equation} | ||
| − | where | + | where <math>x=(x_1,\dots,x_n)</math>, <math>y=(y_1,\dots,y_m)</math>. |
| − | |||
| − | |||
==related items== | ==related items== | ||
| 196번째 줄: | 195번째 줄: | ||
* [[Calogero–Moser-Sutherland model|Calogero–Sutherland model]] | * [[Calogero–Moser-Sutherland model|Calogero–Sutherland model]] | ||
| − | + | ||
==memo== | ==memo== | ||
* http://icmt.illinois.edu/Workshops/WTPCM%20TALKS/bernevig.pdf | * http://icmt.illinois.edu/Workshops/WTPCM%20TALKS/bernevig.pdf | ||
| 202번째 줄: | 201번째 줄: | ||
* http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.160.7557&rep=rep1&type=pdf | * http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.160.7557&rep=rep1&type=pdf | ||
| − | + | ||
==encyclopedia== | ==encyclopedia== | ||
| 213번째 줄: | 212번째 줄: | ||
==articles== | ==articles== | ||
| + | * Charles F. Dunkl, Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials, arXiv:1511.06721[math.RT], November 20 2015, http://arxiv.org/abs/1511.06721v3, 10.3842/SIGMA.2016.033, http://dx.doi.org/10.3842/SIGMA.2016.033, SIGMA 12 (2016), 033, 27 pages | ||
| + | * Piotr Śniady, Top degree of Jack characters and enumeration of maps, http://arxiv.org/abs/1506.06361v2 | ||
| + | * Piotr Śniady, Structure coefficients for Jack characters: approximate factorization property, http://arxiv.org/abs/1603.04268v1 | ||
| + | * Dołęga, Maciej, and Valentin Féray. “Cumulants of Jack Symmetric Functions and <math>b</math>-Conjecture.” arXiv:1601.01501 [math], January 7, 2016. http://arxiv.org/abs/1601.01501. | ||
| + | * Dunkl, Charles F. “Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials.” arXiv:1511.06721 [math], November 20, 2015. http://arxiv.org/abs/1511.06721. | ||
| + | * Śniady, Piotr 2015Top Degree of Jack Characters and Enumeration of Maps. arXiv:1506.06361 [math]. http://arxiv.org/abs/1506.06361, accessed July 11, 2015. | ||
| + | * Lapointe, L., and P. Mathieu. ‘From Jack to Double Jack Polynomials via the Supersymmetric Bridge’. arXiv:1503.09029 [hep-Th, Physics:math-Ph], 31 March 2015. http://arxiv.org/abs/1503.09029. | ||
* Ridout, David, and Simon Wood. “From Jack Polynomials to Minimal Model Spectra.” arXiv:1409.4847 [hep-Th], September 16, 2014. http://arxiv.org/abs/1409.4847. | * Ridout, David, and Simon Wood. “From Jack Polynomials to Minimal Model Spectra.” arXiv:1409.4847 [hep-Th], September 16, 2014. http://arxiv.org/abs/1409.4847. | ||
* Dołęga, Maciej, and Valentin Féray. “On Kerov Polynomials for Jack Characters.” arXiv:1201.1806 [math], January 9, 2012. http://arxiv.org/abs/1201.1806. | * Dołęga, Maciej, and Valentin Féray. “On Kerov Polynomials for Jack Characters.” arXiv:1201.1806 [math], January 9, 2012. http://arxiv.org/abs/1201.1806. | ||
| 221번째 줄: | 227번째 줄: | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
| + | [[분류:symmetric polynomials]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q6115970 Q6115970] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'jack'}, {'LEMMA': 'function'}] | ||
2021년 2월 17일 (수) 02:57 기준 최신판
introduction
It has been known since the early 1970s \cite{Calogero69,Sutherland71, Sutherland72} that \eqref{Cp} with \(\beta=2\gamma\) --- to be denoted \(\exp(-2\gamma W)\) in analogy with \eqref{Fza} --- is the absolute value squared of the ground-state wave function for the Schr\"odinger operator \label{CSpage} \[ H = - \sum_{i=1}^n \frac{\partial^2}{\partial \theta_i^2} + \frac{1}{2}\gamma(\gamma-1)\sum_{1 \le i < j \le n} \frac{1}{\sin^2\bigl(\tfrac{1}{2}(\theta_i-\theta_j)\bigr)}. \] This operator, known as the Calogero--Sutherland Hamiltonian, describes a system of \(n\) identical quantum particles on the unit circle, with \(\theta_i\in[0,2\pi)\) for \(1\leq i\leq n\) the (angular) positions of the particles. The interaction between the particles is described by a \(1/r^2\) two-body potential, \(2|\sin((\theta_i - \theta_j)/2)|\) being the cord-length between particles located at \(\theta_i\) and \(\theta_j\).
B.~Sutherland \cite{Sutherland71} showed that the eigenvalue \(E_0\) corresponding to the ground-state wave function is given by \(E_0=n(n^2-1)\gamma^2/12\). Subsequently he showed \cite{Sutherland72} that the conjugated operator \begin{equation}\label{CO} e^{\gamma W} (H - E_0) e^{-\gamma W} = \sum_{i=1}^n \Bigl(x_i \frac{\partial}{\partial x_i} \Bigr)^2 + 2\gamma \sum_{\substack{i,j=1 \\ i\neq j}}^n \frac{x_i + x_j}{x_i - x_j} \, \frac{\partial}{\partial x_i}, \end{equation} where \(x_j := \exp(i \theta_j)\), admits a complete set of symmetric polynomial eigenfunctions \(P_{\lambda}^{(1/\gamma)}(x)\). These polynomials, now referred to as Jack polynomials, depend on \(x=(x_1,\dots,x_n)\) and are indexed by partitions \(\lambda\) of at most \(n\) parts; \(\lambda=(\lambda_1,\dots,\lambda_n)\) with \(\lambda_1\geq \lambda_2\geq \dots\geq \lambda_n\geq 0\). With \(m_{\lambda}\) denoting the monomial symmetric function indexed by \(\lambda\) and \(<\) the dominance ordering on partitions, the Jack polynomials have the structure \begin{equation}\label{stu} P_{\lambda}^{(1/\gamma)}(x) = m_{\lambda}(x) + \sum_{\mu < \lambda} a_{\lambda\mu}\, m_{\mu}(x) \end{equation} for some coefficients \(a_{\lambda\mu}=a_{\lambda\mu}(\gamma)\).
One fundamental property of the Jack polynomials is that they are orthogonal with respect to the inner product \label{pageJack1} \begin{equation}\label{InProd} \langle f,g \rangle_{\gamma}:=\frac{1}{(2\pi)^n} \int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} f(e^{i \theta}) g(e^{-i \theta}) \prod_{1\le i<j\le n} |e^{i\theta_i}-e^{i\theta_j}|^{2\gamma} \, \,d \theta_1 \cdots \,d \theta_n, \end{equation} where \(f(e^{i \theta})= f(e^{i \theta_1},\dots,e^{i \theta_n})\). To state the orthogonality as well as the quadratic norm evaluation let \[ \Poch{b}{\lambda}{\gamma}=\prod_{i\geq 1} (b+(1-i)\gamma)_{\lambda_i} \] with \((b)_n=b(b+1)\cdots(b+n-1)\) a Pochhammer symbol. Also let \(c_{\lambda}(\gamma)\) and \(c_{\lambda}'(\gamma)\) be given by \[ \label{ccp} \begin{align} c_{\lambda}(\gamma)&=\prod_{s\in\lambda}(a(s)+l(s)\gamma+\gamma), \\ c'_{\lambda}(\gamma)&=\prod_{s\in\lambda}(a(s)+l(s)\gamma+1), \end{align} \] where \(a(s)\) and \(l(s)\) are the arm-length and leg-length of the square \(s\) in the diagram of the partition \(\lambda\), and \(|\lambda|\) is the total number of boxes in the diagram of \(\lambda\) \cite{Macdonald95}. Then \begin{equation}\label{OP} \bigl\langle P_{\lambda}^{(1/\gamma)},P_{\mu}^{(1/\gamma)}\bigr\rangle_{\gamma} =\delta_{\lambda\mu} \: \frac{c'_{\lambda}(\gamma)} {\Poch{1+(n-1)\gamma}{\lambda}{\gamma}}\: \frac{\Gamma(1+n\gamma)}{\Gamma^n(1+\gamma)}\: P_{\lambda}^{(1/\gamma)}(1^n), \end{equation} where \(\delta_{\lambda\mu}\) is the Kronecker delta function and \((1^n)\) is shorthand for \((1,1,\dots,1)\). The orthogonality relation is consistent with, but not an immediate consequence of the operator \eqref{CO} being self-adjoint with respect to \eqref{InProd}. The complication is that not all eigenvalues of \eqref{CO} are distinct. This degeneracy can be removed by introducing the mutually commuting Cherednik operators \(\xi_i\) for \(1\leq i\leq n\) \cite{Cherednik91,Dunkl89} \[ \xi_i=1-i+\frac{x_i}{\gamma} \frac{\partial}{\partial x_i} + \sum_{j=1}^{i-1}\frac{x_i}{x_i - x_j}\,(1 - s_{ij}) + \sum_{j=i+1}^n\frac{x_j}{x_i - x_j}\,(1 - s_{ij}), \] where \(s_{ij}\) acts by permutation \(x_i\) and \(x_j\) and \(1\) represents the identity operator. Any symmetric combination of the \(\xi_i\), and in particular \(\prod_{i=1}^n (1-u\xi_i)\), has the Jack polynomials as simultaneous eigenfunctions.
The Cherednik operators can be used to construct the Jack polynomial shift operator --- a special case of the shift operators studied by Heckman and Opdam, and used by the latter to prove the Macdonald integral and constant term conjectures. Properties of the Jack shift operator not only imply \eqref{CNb1} or, equivalently, \eqref{CNb2}, but also the more general quadratic norm evaluation of the Jack polynomials corresponding to \eqref{OP} with \(\lambda=\mu\) \cite{Kakei98}. (For \(\lambda=0\) this yields \eqref{CNb1}.) With \(\Delta(x)\) the Vandermonde product \eqref{VanderM} and \(Y_{\pm}:= \gamma^{n(n-1)/2} \prod_{1\leq i<j\leq n}(\xi_i-\xi_j\mp 1)\), the Jack shift operators are defined by \(G_{+}:=\Delta^{-1} Y_{+}\), \(G_{-} = Y_{-}\Delta\). They have an adjoint type property with respect to the inner product \eqref{InProd}, \begin{equation}\label{fr.1} \langle G_{+}f,g\rangle_{\gamma+1}=\langle f,G_{-} g\rangle_{\gamma}. \end{equation} Also, with \begin{equation}\label{apm} a_{\lambda}^{\pm}(\gamma)=\prod_{1\leq i<j\leq n} (\lambda_i-\lambda_j\pm 1+(j-i\mp 1)\gamma) \end{equation} and \(\delta\) the staircase partition \((n-1,n-2,\dots,1,0)\), the shift operators act on the Jack polynomials as \[\label{fr.2} \begin{align} G_{+} P_{\lambda+\delta}^{(1/\gamma)}&=a_{\lambda}^{+}(\gamma+1) P_{\lambda}^{(1/(\gamma+1))}, \\[2mm] G_{-} P_{\lambda}^{(1/(\gamma+1))}&=a_{\lambda}^{-}(\gamma+1) P_{\lambda+\delta}^{(1/\gamma )}. \end{align} \] It follows from \eqref{fr.1} and \eqref{fr.2} that \[ \bigl\langle P_{\lambda}^{(1/(\gamma+1))},P_{\lambda}^{(1/(\gamma+1))} \bigr\rangle_{\gamma+1}= \frac{a_{\lambda}^{-}(\gamma+1)}{a_{\lambda}^{+}(\gamma+1)}\: \bigl\langle P_{\lambda+\delta}^{(1/\gamma)},P_{\lambda+\delta}^{(1/\gamma)} \bigr\rangle_{\gamma} \] and thus \[ \bigl\langle P_{\lambda}^{(1/(\gamma+k))},P_{\lambda}^{(1/(\gamma+k))} \bigr\rangle_{\gamma+k}= \bigl\langle P_{\lambda+k\delta}^{(1/\gamma)},P_{\lambda+k\delta}^{(1/\gamma)} \bigr\rangle_{\gamma}\: \prod_{j=1}^{k-1} \frac{a_{\lambda+j\delta}^{-}(\gamma+k-j)}{a_{\lambda+j\delta}^{+}(\gamma+k-j)}. \] Taking \(\gamma=0\), using that \(P_{\lambda}^{(\infty)}=m_{\lambda}\) (the monomial symmetric function) and \[ \bigl\langle m_{\mu},m_{\mu} \bigr\rangle_0= \text{CT}\Bigl( m_{\mu}(x)m_{\mu}(x^{-1})\Bigr)= m_{\mu}(1^n) \] which is \(n!\) for \(\mu=\lambda+k\delta\), it follows that for nonnegative integer \(k\) \label{pageJack2} \begin{equation}\label{fr.3} \bigl\langle P_{\lambda}^{(1/k)},P_{\lambda}^{(1/k)}\bigr\rangle_k =n!\prod_{j=0}^{k-1} \frac{a_{\lambda+jk}^{-}(k-j)} {a_{\lambda+jk}^{+}(k-j)}. \end{equation} Using the evaluation formula \cite{Stanley89} \begin{equation}\label{ef} P_{\lambda}^{(1/\gamma)}(1^n)=\frac{[n\gamma]_{\lambda}^{(\gamma)}} {c_{\lambda}(\gamma)} \end{equation} and the definitions \eqref{ccp} and \eqref{apm} it is now a straightforward exercise to verify that for \(\gamma=k\) \eqref{OP} coincides with \eqref{fr.3}. Analytic continuation off the integers is then required to establish \eqref{OP} for all \(\Re(\gamma)>-1/n\).
\medskip
A further fundamental property of the Jack polynomials is R.P.~Stanley's Cauchy identity \cite{Stanley89} \begin{equation}\label{CaP} \sum_{\lambda} \frac{c_{\lambda}(\gamma)}{c'_{\lambda}(\gamma)} \, P_{\lambda}^{(1/\gamma)}(x) P_{\lambda}^{(1/\gamma)}(y)= \prod_{i=1}^n\prod_{j=1}^m (1-x_i y_j)^{-\gamma}, \end{equation} where \(x=(x_1,\dots,x_n)\), \(y=(y_1,\dots,y_m)\).
memo
- http://icmt.illinois.edu/Workshops/WTPCM%20TALKS/bernevig.pdf
- http://www-users.math.umd.edu/~harryt/papers/schurrev.pdf
- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.160.7557&rep=rep1&type=pdf
encyclopedia
- http://en.wikipedia.org/wiki/Jack_polynomial
- http://en.wikipedia.org/wiki/Schur_polynomial
- Schur functions in algebraic combinatorics
- http://planetmath.org/encyclopedia/SchurPolynomial.html
articles
- Charles F. Dunkl, Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials, arXiv:1511.06721[math.RT], November 20 2015, http://arxiv.org/abs/1511.06721v3, 10.3842/SIGMA.2016.033, http://dx.doi.org/10.3842/SIGMA.2016.033, SIGMA 12 (2016), 033, 27 pages
- Piotr Śniady, Top degree of Jack characters and enumeration of maps, http://arxiv.org/abs/1506.06361v2
- Piotr Śniady, Structure coefficients for Jack characters: approximate factorization property, http://arxiv.org/abs/1603.04268v1
- Dołęga, Maciej, and Valentin Féray. “Cumulants of Jack Symmetric Functions and \(b\)-Conjecture.” arXiv:1601.01501 [math], January 7, 2016. http://arxiv.org/abs/1601.01501.
- Dunkl, Charles F. “Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials.” arXiv:1511.06721 [math], November 20, 2015. http://arxiv.org/abs/1511.06721.
- Śniady, Piotr 2015Top Degree of Jack Characters and Enumeration of Maps. arXiv:1506.06361 [math]. http://arxiv.org/abs/1506.06361, accessed July 11, 2015.
- Lapointe, L., and P. Mathieu. ‘From Jack to Double Jack Polynomials via the Supersymmetric Bridge’. arXiv:1503.09029 [hep-Th, Physics:math-Ph], 31 March 2015. http://arxiv.org/abs/1503.09029.
- Ridout, David, and Simon Wood. “From Jack Polynomials to Minimal Model Spectra.” arXiv:1409.4847 [hep-Th], September 16, 2014. http://arxiv.org/abs/1409.4847.
- Dołęga, Maciej, and Valentin Féray. “On Kerov Polynomials for Jack Characters.” arXiv:1201.1806 [math], January 9, 2012. http://arxiv.org/abs/1201.1806.
- Sahi, Siddhartha. “A New Scalar Product for Nonsymmetric Jack Polynomials.” International Mathematics Research Notices 1996, no. 20 (January 1, 1996): 997–1004. doi:10.1155/S107379289600061X.
메타데이터
위키데이터
- ID : Q6115970
Spacy 패턴 목록
- [{'LOWER': 'jack'}, {'LEMMA': 'function'}]