Jack polynomials

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 \(x=(x_1,\dots,x_n)\), \(y=(y_1,\dots,y_m)\).

related items

memo

encyclopedia

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.

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[{'LOWER': 'jack'}, {'LEMMA': 'function'}]