Vector-valued Jack Polynomials and Wavefunctions on the Torus

TL;DR

This paper introduces a new class of eigenfunctions for the Calogero-Sutherland model using vector-valued Jack polynomials, expanding the mathematical framework for quantum many-body systems on the torus.

Contribution

It develops a novel construction of eigenfunctions employing generalized Jack polynomials valued in symmetric group modules, applicable to all irreducible representations.

Findings

Constructed eigenfunctions using matrix solutions of differential systems.

Generated symmetric probability densities on the N-torus from these eigenfunctions.

Extended the theory of Jack polynomials to vector-valued cases for quantum models.

Abstract

The Hamiltonian of the quantum Calogero-Sutherland model of $N$ identical particles on the circle with $1/r^{2}$ interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the $N$-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author.

Equations

$$\mathcal{H}$$

$$=\sum _{i=1}^N{\left(x_i\frac{\partial }{\partial x_i}\right)}^2-2\kappa \sum _{1\le i<j\le N}\frac{x_i{x}_j\left(\kappa -1\right)}{{\left(x_i-x_j\right)}^2},$$

$${\psi }_0\left(x\right)=\prod _{1\le i<j\le N}{\left(-\frac{{\left(x_i-x_j\right)}^2}{x_i{x}_j}\right)}^{\kappa /2}$$

$$\frac{\partial }{\partial x_i}{\psi }_0\left(x\right)=\kappa {\psi }_0\left(x\right)\left\{\sum _{j\ne i}\frac{1}{x_i-x_j}-\frac{N-1}{2x_i}\right\},1\le i\le N;$$

$${\left(xw\right)}^{\alpha }=\prod _{i=1}^N{x}_{w\left(i\right)}^{{\alpha }_i}=\prod _{j=1}^N{x}_j^{{\alpha }_{w^{-1}\left(j\right)}},$$

$$s_i{s}_j$$

$$s_i{s}_{i+1}{s}_i$$

$$h\left(i,j\right)$$

$$h_{\tau }$$

$${⟨T,T^{\prime }⟩}_0:={\delta }_{T,T^{\prime }}\times \prod _{\begin{array}{c}1\le i<j\le N,\\ c\left(i,T\right)\le c\left(j,T\right)-2\end{array}}\left(1-\frac{1}{{\left(c\left(i,T\right)-c\left(j,T\right)\right)}^2}\right),T,T^{\prime }\in \mathcal{Y}\left(\tau \right).$$

$$w\left(x^{\alpha }\otimes T\right)$$

$$wp\left(x\right)$$

$$\frac{1}{{⟨T,T⟩}_0}{p}_T\left(x\right)\otimes T+\frac{1}{{⟨T^{\left(i\right)},T^{\left(i\right)}⟩}_0}{p}_{T^{\left(i\right)}}\left(x\right)\otimes T^{\left(i\right)}$$

$$=\frac{1}{{⟨T,T⟩}_0}{p}_T\left(x{s}_i\right)\otimes \tau \left(s_i\right)T+\frac{1}{{⟨T^{\left(i\right)},T^{\left(i\right)}⟩}_0}{p}_{T^{\left(i\right)}}\left(x{s}_i\right)\otimes \tau \left(s_i\right)T^{\left(i\right)}$$

$$=\frac{1}{{⟨T,T⟩}_0}\left\{p_T\left(x{s}_i\right)\otimes \left(T^{\left(i\right)}+bT\right)+\frac{1}{1-b^2}{p}_{T^{\left(i\right)}}\left(x{s}_i\right)\otimes \left(\left(1-b^2\right)T-b{T}^{\left(i\right)}\right)\right\}$$

$$=\frac{1}{{⟨T,T⟩}_0}\left\{\left(b{p}_T\left(x{s}_i\right)+p_{T^{\left(i\right)}}\left(x{s}_i\right)\right)\otimes T+\left(\left(1-b^2\right)p_T\left(x{s}_i\right)-b{p}_{T^{\left(i\right)}}\left(x{s}_i\right)\right)\otimes T^{\left(i\right)}\right\}.$$

$${\mathcal{D}}_{i}p\left(x\right)$$

$${\mathcal{U}}_{i}p\left(x\right)$$

$${\mathcal{D}}_i{\mathcal{D}}_j$$

$$w{\mathcal{D}}_i$$

$$s_i{\mathcal{U}}_i{s}_i$$

$${\mathcal{U}}_i={\mathcal{D}}_i{x}_i-\kappa \sum _{j<i}\left(i,j\right)=x_i{\mathcal{D}}_i+1+\kappa \sum _{j>i}\left(i,j\right)=x_i{\mathcal{D}}_i+1+\kappa {\omega }_i.$$

$$s_i\left({\mathcal{U}}_i+{\mathcal{U}}_{i+1}\right)s_i$$

$$s_i{\mathcal{U}}_i{\mathcal{U}}_{i+1}{s}_i$$

$${\partial }_{i}L\left(x\right)$$

$$\gamma$$

$${\mathbb{T}}^N$$

$$\mathrm{d}m\left(x\right)$$

$$\det L\left(x\right)=c\prod _{1\le i<j\le N}{\left(-\frac{{\left(x_i-x_j\right)}^2}{x_i{x}_j}\right)}^{\kappa \lambda /2},\lambda :=\frac{\gamma n_{\tau }}{2\left(N-1\right)}=\mathrm{tr}\left(\tau \left(1,2\right)\right),$$

$$L\left(x\right):=L\left(x{w}_x^{-1}\right)\tau \left(w_x\right).$$

$$M\left(w,x\right):=\tau {\left(w_0\right)}^{1-w_{x}w\left(1\right)}.$$

$$M\left(I,x\right)$$

$$M\left(w_1{w}_2,x\right)$$

$$L\left(xw\right)$$

$$L\left(x\right)wp\left(x\right)$$

$$=L\left(x\right)\tau \left(w\right){\left\{M\left(w,x\right)L\left(x\right)\tau \left(w\right)\right\}}^{-1}f\left(xw\right)$$

$$=M{\left(w,x\right)}^{-1}f\left(xw\right),$$

$${\sigma }^M\left(w\right)f\left(x\right)=M{\left(w,x\right)}^{-1}f\left(xw\right).$$

$${\sigma }^M\left(w_1\right){\sigma }^M\left(w_2\right)f\left(x\right)$$

Full text

1 Introduction

The quantum Calogero-Sutherland model for $N$ identical particles with $1/r^{2}$ interactions on the unit circle has the Hamiltonian

[TABLE]

where $x_{j}=e^{\mathrm{i}\theta_{j}}$ and $-\pi<\theta_{j}\leq\pi$ for $1\leq j\leq N$. The time-independent Schrödinger equation $\mathcal{H}\psi=E\psi$ has solutions expressible as the product of the base-state

[TABLE]

with a Jack polynomial (Lapointe and Vinet [8], Awata [1]). The base-state is a solution of the first-order linear differential system

[TABLE]

and $\mathcal{H}\psi_{0}=\frac{1}{12}\kappa^{2}N\left(N^{2}-1\right)\psi_{0}$. The theory of Jack polynomials has been generalized to polynomials taking values in modules of the symmetric group (Griffeth [6]). In this paper the Hamiltonian $\mathcal{H}$ will be interpreted in that context. The base state $\psi_{0}$ is replaced by a matrix function satisfying an analogous differential system and the generalized wavefunctions are vector-valued. Nevertheless for an interval of parameter values depending on the module the wavefunctions do give rise to symmetric probability density functions on the torus. The interval is symmetric about $\kappa=0$ hence this is qualitatively different from the usual scalar case where $\kappa$ is unbounded above.

Section 2 is a brief overview of representation theory for the symmetric groups, and the commutative set of operators on polynomials of which the nonsymmetric Jack polynomials are simultaneous eigenfunctions. Section 3 concerns the first-order linear differential system defining the basic matrix function needed to map the polynomials to eigenfunctions of the Hamiltonian modified with twisted exchange operators. In Section 4 there is a description of the Hermitian form related to integration of vector-valued polynomials on the torus, and the Yang-Baxter graph technique for constructing the nonsymmetric Jack polynomials. Section 5 presents the adaptation of the method of Baker and Forrester [2] to form symmetric Jack polynomials from the nonsymmetric polynomials; the analysis involves tableaux with certain properties. Also this section contains the formulae for the squared norms of the Jack polynomials. Then Section 6 uses the vector-valued Jack polynomials and the matrix function from Section 3 to construct vector-valued eigenfunctions of the Hamiltonian $\mathcal{H}$ and the associated probability density. Also the Jack polynomial of minimal degree is described, and finally there is a brief description of the matrix function in the case of the two-dimensional representation of $\mathcal{S}_{4}$.

2 The generalized Jack polynomials and associated

operators

The symmetric group $\mathcal{S}_{N}$, the set of permutations of $\left\{1,2,\ldots,N\right\}$, acts on $\mathbb{C}^{N}$ by permutation of coordinates. For $\alpha\in\mathbb{Z}^{N}$ the norm is $\left|\alpha\right|:=\sum_{i=1}^{N}\left|\alpha_{i}\right|$ and the monomial is $x^{\alpha}:=\prod_{i=1}^{N}x_{i}^{\alpha_{i}}$. Denote $\mathbb{N}_{0}:=\left\{0,1,2,\ldots\right\}$. The space of polynomials $\mathcal{P}:=\mathrm{span}_{\mathbb{C}}\left\{x^{\alpha}:\alpha\in\mathbb{N}_{0}^{N}\right\}$. Elements of $\mathrm{span}_{\mathbb{C}}\left\{x^{\alpha}:\alpha\in\mathbb{Z}^{N}\right\}$ are called Laurent polynomials. The action of $\mathcal{S}_{N}$ is extended to polynomials by $wp\left(x\right)=p\left(xw\right)$ where $\left(xw\right)_{i}=x_{w\left(i\right)}$ (consider $x$ as a row vector and $w$ as a permutation matrix, $\left[w\right]_{ij}=\delta_{i,w\left(j\right)}$, then $xw=x\left[w\right]$). This is a representation of $\mathcal{S}_{N}$, that is, $w_{1}\left(w_{2}p\right)\left(x\right)=\left(w_{2}p\right)\left(xw_{1}\right)=p\left(xw_{1}w_{2}\right)=\left(w_{1}w_{2}\right)p\left(x\right)$ for all $w_{1},w_{2}\in\mathcal{S}_{N}$.

Furthermore $\mathcal{S}_{N}$ is generated by reflections in the mirrors $\left\{x:x_{i}=x_{j}\right\}$ for $1\leq i<j\leq N$. These are *transpositions, *denoted by $\left(i,j\right)$, interchanging $x_{i}$ and $x_{j}$. Define the $\mathcal{S}_{N}$-action on $\alpha\in\mathbb{Z}^{N}$ so that $\left(xw\right)^{\alpha}=x^{w\alpha}$

[TABLE]

that is $\left(w\alpha\right)_{i}=\alpha_{w^{-1}\left(i\right)}$ (consider $\alpha$ as a column vector, then $w\alpha=\left[w\right]\alpha$).

The simple reflections $s_{i}:=\left(i,i+1\right)$, $1\leq i<N$, suffice to generate $\mathcal{S}_{N}$. They are the key devices for applying inductive methods, and satisfy the braid relations:

[TABLE]

We consider the situation where the group $\mathcal{S}_{N}$ acts on the range as well as on the domain of the polynomials. We use vector spaces, called $\mathcal{S}_{N}$-modules, on which $\mathcal{S}_{N}$ has an irreducible unitary (orthogonal) representation$:\tau:\mathcal{S}_{N}\rightarrow O_{m}\left(\mathbb{R}\right)$ ($\tau\left(w\right)^{-1}=\tau\left(w^{-1}\right)=\tau\left(w\right)^{T}$). See James and Kerber [7] for representation theory and a modern discussion of Young’s methods.

Denote the set of partitions $\mathbb{N}_{0}^{N,+}=\left\{\lambda\in\mathbb{N}_{0}^{N}:\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\right\}$. We identify $\tau$ with a partition of $N$ given the same label, that is $\tau\in\mathbb{N}_{0}^{N,+}$ and $\left|\tau\right|=N$. The length of $\tau$ is $\ell\left(\tau\right)=\max\left\{i:\tau_{i}>0\right\}$. There is a Ferrers diagram of shape $\tau$ (also given the same label), with boxes at points $\left(i,j\right)$ with $1\leq i\leq\ell\left(\tau\right)$ and $1\leq j\leq\tau_{i}$. A tableau of shape $\tau$ is a filling of the boxes with numbers, and a reverse standard Young tableau (RSYT) is a filling with the numbers $\left\{1,2,\ldots,N\right\}$ so that the entries decrease in each row and each column. We require $\dim V_{\tau}\geq 2$, thus excluding the one-dimensional representations corresponding to one-row $\left(N\right)$ or one-column $\left(1,1,\ldots,1\right)$ partitions (the trivial and determinant representations, respectively). The hook-length of the node $\left(i,j\right)\in\tau$ is

[TABLE]

and $h_{\tau}$ is the maximum hook-length of $\tau$. Denote the set of RSYT’s of shape $\tau$ by $\mathcal{Y}\left(\tau\right)$ and let $V_{\tau}=\mathrm{span}_{\mathbb{C}}\left\{T:T\in\mathcal{Y}\left(\tau\right)\right\}$ with orthogonal basis $\mathcal{Y}\left(\tau\right)$. The formulae for the action of $s_{i}$ on $\mathcal{Y}\left(\tau\right)$ are described in Proposition 1 below. The hook-length formula is $\#\mathcal{Y}\left(\tau\right)=N!/\prod\limits_{\left(i,j\right)\in\tau}h\left(i,j\right)$. Set $n_{\tau}:=\dim V_{\tau}=\#\mathcal{Y}\left(\tau\right)$. For $1\leq i\leq N$ and $T\in\mathcal{Y}\left(\tau\right)$ the entry $i$ is at coordinates $\left(\mathrm{rw}\left(i,T\right),\mathrm{cm}\left(i,T\right)\right)$ and the content is $c\left(i,T\right):=\mathrm{cm}\left(i,T\right)-\mathrm{rw}\left(i,T\right)$. Each $T\in\mathcal{Y}\left(\tau\right)$ is uniquely determined by its content vector $\left[c\left(i,T\right)\right]_{i=1}^{N}$. Let $S_{1}\left(\tau\right):=\sum_{i=1}^{N}c\left(i,T\right)$ (this sum depends only on $\tau$) and $\gamma:=S_{1}\left(\tau\right)/N$. The $\mathcal{S}_{N}$-invariant inner product on $V_{\tau}$ is defined by

[TABLE]

It is unique up to multiplication by a constant.

The Jucys-Murphy elements $\omega_{i}:=\sum\limits_{j=i+1}^{N}\left(i,j\right)$ satisfy $\sum\limits_{j=i+1}^{N}\tau\left(i,j\right)T=c\left(i,T\right)T$ and thus the central element $\sum\limits_{1\leq i<j\leq N}\left(i,j\right)$ (in the group algebra $\mathbb{R}\mathcal{S}_{N}$) satisfies $\sum\limits_{1\leq i<j\leq N}\tau\left(i,j\right)T=S_{1}\left(\tau\right)T$ for each $T\in\mathcal{Y}\left(\tau\right)$. We abbreviate $\tau\left(\left(i,j\right)\right)$ to $\tau\left(i,j\right).$

The generalized Jack polynomials are elements of $\mathcal{P}_{\tau}=\mathcal{P}\otimes V_{\tau}$, the space of $V_{\tau}$-valued polynomials, which is equipped with the $\mathcal{S}_{N}$ action:

[TABLE]

extended by linearity. A symmetric polynomial $p$ satisfies $wp=p$, that is, $p\left(xw\right)=\tau\left(w\right)^{-1}p\left(x\right)$ for all $w\in\mathcal{S}_{N}$. . The following describes the transformation rules for $\tau\left(s_{i}\right)$ acting on $\mathcal{Y}\left(\tau\right)$ and on symmetric polynomials.

Proposition 1

Suppose $p\in\mathcal{P}_{\tau}$ is symmetric, $1\leq i<N$, and $T\in\mathcal{Y}\left(\tau\right)$. Express $p\left(x\right)=\sum\limits_{T^{\prime}\in\mathcal{Y}\left(\tau\right)}\frac{1}{\left\langle T^{\prime},T^{\prime}\right\rangle_{0}}p_{T^{\prime}}\left(x\right)\otimes T^{\prime}.$ If $c\left(i,T\right)=c\left(i+1,T\right)+1$ then $\tau\left(s_{i}\right)T=T$ and $s_{i}p_{T}=p_{T}$; if $c\left(i,T\right)=c\left(i+1,T\right)-1$ then $\tau\left(s_{i}\right)T=-T$ and $s_{i}p_{T}=-p_{T}$; if $c\left(i,T\right)-c\left(i+1,T\right)\geq 2$ and $T^{\left(i\right)}$ is $T$ with $i,i+1$ interchanged then $\tau\left(s_{i}\right)T=T^{\left(i\right)}+bT$ and $s_{i}p_{T}=p_{T^{\left(i\right)}}+bp_{T}$, where $b=\frac{1}{c\left(i,T\right)-c\left(i+1,T\right)}$.

Proof. The transformation properties of $p_{T}$ follow from $s_{i}\left(p_{T}\left(x\right)\otimes T\right)=p\left(xs_{i}\right)\otimes\tau\left(s_{i}\right)T$. The first case is when $\mathrm{rw}\left(i,T\right)=\mathrm{rw}\left(i+1,T\right)$ and $\tau\left(s_{i}\right)T=T$; the second case is when $\mathrm{cm}\left(i,T\right)=\mathrm{cm}\left(i+1,T\right)$ and $\tau\left(s_{i}\right)T=-T$. In the case $c\left(i,T\right)-c\left(i+1,T\right)\geq 2$ the relations $\tau\left(s_{i}\right)T^{\left(i\right)}=\left(1-b^{2}\right)T-bT^{\left(i\right)}$ and $\left\langle T^{\left(i\right)},T^{\left(i\right)}\right\rangle_{0}=\left(1-b^{2}\right)\left\langle T,T\right\rangle_{0}$ hold. By hypothesis

[TABLE]

Replace $x$ by $xs_{i}$ and conclude that $s_{i}p_{T}=p_{T^{\left(i\right)}}+bp_{T}$ and $s_{i}p_{T^{\left(i\right)}}=\left(1-b^{2}\right)p_{T}-bp_{T^{\left(i\right)}}$.

The polynomials $p_{T}$ can be derived from $p_{T_{0}}$ where $T_{0}$ is the root RSYT (with $N,N-1,\ldots$ entered column by column), but determining which polynomials can serve as $p_{T_{0}}$ is nontrivial in general.

There is a parameter $\kappa\in\mathbb{R}$ (in general, $\kappa$ could be transcendental).

Definition 2

The Dunkl and Cherednik-Dunkl operators are ($1\leq i\leq N,p\in\mathcal{P}_{\tau}$)

[TABLE]

The commutation relations analogous to the scalar case hold:

[TABLE]

The commutation properties for the $\mathcal{U}_{i}$ and $s_{j}$ are derived as follows:

if $j>i$ then $s_{j}$ commutes with each term in $\mathcal{U}_{i}$, 2. 2.

if $j<i-1$ then $\mathcal{D}_{i}x_{i}s_{j}-\kappa\sum_{k<i}\left(k,i\right)s_{j}=s_{j}\mathcal{D}_{i}x_{i}-\kappa\sum_{k<i}s_{j}\left(k,i\right)$ because $\left(k,i\right)s_{j}=s_{j}\left(k,i\right)$ unless $k=j$ or $j+1$ and then $\left\{\left(i,j\right)+\left(i,j+1\right)\right\}s_{j}=s_{j}\left\{\left(i,j+1\right)+\left(i,j\right)\right\}$, 3. 3.

if $j=i$ then $s_{i}\mathcal{U}_{i}s_{i}=\mathcal{D}_{i+1}x_{i+1}-\kappa\sum_{k<i}s_{i}\left(k,i\right)s_{i}=\mathcal{D}_{i+1}x_{i+1}-\kappa\sum_{k<i}\left(k,i+1\right)=\mathcal{U}_{i+1}+\kappa\left(i,i+1\right)$; the relations $\mathcal{U}_{i}s_{i}=s_{i}\mathcal{U}_{i+1}+\kappa,~{}\mathcal{U}_{i+1}s_{i}=s_{i}\mathcal{U}_{i}-\kappa$ follow from right and left multiplication by $s_{i}$.

From the commutation $\mathcal{D}_{i}x_{i}-x_{i}\mathcal{D}_{i}=1+\kappa\sum_{j\neq i}\left(i,j\right)$ we obtain

[TABLE]

Proposition 3

If $q\left(x_{1},x_{2},\ldots,x_{N}\right)$ is a symmetric polynomial then $q\left(\mathcal{U}_{1},\mathcal{U}_{2},\ldots,\mathcal{U}_{N}\right)$ commutes with each $w\in\mathcal{S}_{N}$, as an operator on $\mathcal{P}_{\tau}$.

Proof. It suffices to prove the commutativity for each $s_{i}$ with $1\leq i<N$ and each elementary symmetric polynomial in $\left\{\mathcal{U}_{i}\right\}$, that is, for $\prod\limits_{j=1}^{N}\left(1+t\mathcal{U}_{j}\right)$. By the above formulae it suffices to show $s_{i}$ commutes with $\left(1+t\mathcal{U}_{i}\right)\left(1+t\mathcal{U}_{i+1}\right)=1+t\left(\mathcal{U}_{i}+\mathcal{U}_{i+1}\right)+t^{2}\mathcal{U}_{i}\mathcal{U}_{i+1}$. Indeed

[TABLE]

The nonsymmetric (vector-valued) Jack polynomials (NSJP) are defined to be simultaneous eigenfunctions of the commuting set $\left\{\mathcal{U}_{i}:1\leq i\leq N\right\}$. The symmetric vector-valued Jack polynomials are simultaneous eigenfunctions of the symmetric polynomials in $\left\{\mathcal{U}_{i}\right\}$. If $p$ is a NSJP then the sum $\sum\limits_{w\in\mathcal{S}_{N}}wp$ is either a scalar multiple of a symmetric Jack polynomial or zero. The details are presented in Section 5.

3 The matrix analogue of the base state

This is a summary of the pertinent results from [4]. Vectors and matrices throughout are of size $n_{\tau}$ and $n_{\tau}\times n_{\tau}$ and are expressed with respect to the orthonormal basis $\left\{\left\langle T,T\right\rangle_{0}^{-1/2}T:T\in\mathcal{Y}\left(\tau\right)\right\}$. With $\partial_{i}:=\frac{\partial}{\partial x_{i}}$ for $1\leq i\leq N$ the differential system for the matrix function $L$ is

[TABLE]

The effect of the term $\frac{\gamma}{x_{i}}I$ is to make $L\left(x\right)$ homogeneous of degree zero, that is, $\sum_{i=1}^{N}x_{i}\partial_{i}L\left(x\right)=0$. The differential system is defined on $\mathbb{C}_{reg}^{N}:=\mathbb{C}_{\times}^{N}\backslash\bigcup\limits_{1\leq i<j\leq N}\left\{x:x_{i}=x_{j}\right\}$ (where $\mathbb{C}_{\times}:=\mathbb{C}\backslash\left\{0\right\}$), and it is Frobenius integrable and analytic, thus any local solution can be continued analytically to any point in $\mathbb{C}_{reg}^{N}$. The equation is a modified version of the Knizhnik-Zamolodchikov equation. For the trivial representation $\tau=\left(N\right)$ the solution $L\left(x\right)=\psi_{0}\left(x\right)$ (up to scalar multiplication) because $\tau\left(i,j\right)=I$ and $\gamma=\frac{N-1}{2}$. The notations for the torus and its surface measure in terms of polar coordinates are

[TABLE]

Let $\mathbb{T}_{reg}^{N}:=\mathbb{T}^{N}\cap\mathbb{C}_{reg}^{N}$, then $\mathbb{T}_{reg}^{N}$ has $\left(N-1\right)!$ connected components and each component is homotopic to a circle; if $x$ is in some component then so is $ux=\left(ux_{1},\ldots,ux_{N}\right)$ for each $u\in\mathbb{T}$.

Definition 4

Let $x_{0}:=\left(1,e^{2\pi\mathrm{i}/N},e^{4\pi\mathrm{i}/N},\ldots,e^{2\left(N-1\right)\pi\mathrm{i}/N}\right)$ and denote the connected component of $\mathbb{T}_{reg}^{N}$ containing $x_{0}$ by $\mathcal{C}_{0}$, called the fundamental chamber.

Thus $\mathcal{C}_{0}$ is the set consisting of $\left(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\right)$ with $\theta_{1}<\theta_{2}<\cdots<\theta_{N}<\theta_{1}+2\pi$. The homogeneity $L\left(ux\right)=L\left(x\right)$ for $\left|u\right|=1$ shows that $L\left(x\right)$ has a well-defined analytic continuation to all of $\mathcal{C}_{0}$ starting from $x_{0}$. Let $w_{0}:=\left(1,2,3,\ldots N\right)=\left(12\right)\left(23\right)\cdots\left(N-1,N\right)$, an $N$-cycle, and let $\left\langle w_{0}\right\rangle$ denote the cyclic group generated by $w_{0}$. There are two components of $\mathbb{T}_{reg}^{N}$ which are set-wise invariant under $\left\langle w_{0}\right\rangle$ namely $\mathcal{C}_{0}$ and the reverse $\left\{\theta_{N}<\theta_{N-1}<\ldots<\theta_{1}<\theta_{N}+2\pi\right\}$. Indeed $\left\langle w_{0}\right\rangle$ is the stabilizer of $\mathcal{C}_{0}$ as a subgroup of $\mathcal{S}_{N}$. A list of properties of $L\left(x\right)$ (from [4]):

If $L\left(x\right)$ is a solution of (5) in some connected open subset $U$ of $\mathbb{C}_{reg}^{N}$ then $L\left(xw\right)\tau\left(w\right)^{-1}$ is a solution in $Uw^{-1}$; 2. 2.

If $L\left(x_{0}\right)$ is nonsingular then $L$ is nonsingular on all of $\mathcal{C}_{0}$; this follows from

[TABLE] 3. 3.

Suppose $L\left(x\right)$ is normalized by $L\left(x_{0}\right)=I$ then $L\left(xw_{0}^{m}\right)=\tau\left(w_{0}\right)^{-m}L\left(x\right)\tau\left(w_{0}\right)^{m}$ for all $x\in\mathcal{C}_{0}$ and $m\in\mathbb{Z}$;

For $w\in\mathcal{S}_{N}$ and for $x\in\mathbb{T}_{reg}^{N}$ define $w_{x}\in\mathcal{S}_{N}$ such that $xw_{x}^{-1}\in\mathcal{C}_{0}$ and $w_{x}\left(1\right)=1$; then $w_{x}$ is uniquely defined and is constant on connected components. Then define $L\left(x\right)$ on the other connected components of $\mathbb{T}_{reg}^{N}$ by

[TABLE]

In order to derive a formula for the relation of $L\left(xw\right)\tau\left(w\right)^{-1}$ to $L\left(x\right)$ we need a twist: for $w\in\mathcal{S}_{N}$ and for $x\in\mathbb{T}_{reg}^{N}$ define

[TABLE]

Henceforth the assumption $L\left(x_{0}\right)=I$ is relaxed to $L\left(x_{0}\right)$ commuting with $\tau\left(w_{0}\right)$ and being nonsingular (so $L\left(xw_{0}^{m}\right)=\tau\left(w_{0}\right)^{-m}L\left(x\right)\tau\left(w_{0}\right)^{m}$ still holds for $x\in\mathcal{C}_{0}$). Then $M\left(w,x\right)$ and $L\left(x\right)$ have the following properties ($x\in\mathbb{T}_{reg}^{N}$):

[TABLE]

With the goal of analyzing vector functions of the form $f\left(x\right)=L\left(x\right)p\left(x\right)$ where $p\in\mathcal{P}_{\tau}$ consider

[TABLE]

in other words $L\left(x\right)wL\left(x\right)^{-1}f\left(x\right)=M\left(w,x\right)^{-1}f\left(xw\right)$. Accordingly define a twisted action of $\mathcal{S}_{N}$ on vector-valued functions $f\left(x\right)$ (defined on $\mathbb{T}_{reg}^{N}$) by

[TABLE]

Proposition 5

Suppose $w\in\mathcal{S}_{N}$ then $L\left(x\right)wL\left(x\right)^{-1}=\sigma^{M}\left(w\right)$ and $\sigma^{M}$ is a representation of $\mathcal{S}_{N}$.

Proof. Using formula (7) let $g\left(x\right)=\sigma^{M}\left(w_{2}\right)f\left(x\right)=M\left(w_{2},x\right)^{-1}f\left(xw_{2}\right)$, then

[TABLE]

If $p\left(x\right)$ is symmetric ($\tau\left(w\right)p\left(xw\right)=p\left(x\right)$) then

[TABLE]

This is crucial in the sequel where the operator $L\left(x\right)\sum_{i=1}^{N}\left(\mathcal{U}_{i}-1-\kappa\gamma\right)^{2}L\left(x\right)^{-1}$ is related to the Hamiltonian $\mathcal{H}$.

Proposition 6

For $1\leq i\leq N$

[TABLE]

Proof. Write the differential system as $\partial_{i}L\left(x\right)=\kappa L\left(x\right)A_{i}\left(x\right)$ then

[TABLE]

thus $\partial_{i}L\left(x\right)^{-1}=-\kappa A_{i}\left(x\right)L\left(x\right)^{-1}$. Next by formula (8) $L\left(x\right)\left(i,j\right)L\left(x\right)^{-1}=\sigma^{M}\left(i,j\right)$. For the other term in $\mathcal{U}_{i}=x_{i}\mathcal{D}_{i}+1+\kappa\omega_{i}$ (formula (4)) we obtain $L\left(x\right)\omega_{i}L\left(x\right)^{-1}=\sum_{j>i}\sigma^{M}\left(i,j\right)$. Consider

[TABLE]

Thus

[TABLE]

We will use an elementary double sum formula: suppose $g\left(i,j\right)$ is a function defined on all pairs $\left(i,j\right)$ with $1\leq i,j\leq N$ then

[TABLE]

Corollary 7

$\sum_{i=1}^{N}\mathcal{U}_{i}=\sum_{i=1}^{N}x_{i}\partial_{i}+N+\kappa S_{1}.$**

Proof. From formula (9)

[TABLE]

The double sum is of the form (10) and for $i<j$ one obtains $g\left(i,j\right)+g\left(j,i\right)=\frac{x_{i}}{x_{i}-x_{j}}\sigma^{M}\left(i,j\right)+\frac{x_{i}}{x_{j}-x_{i}}\sigma^{M}\left(j,i\right)=0$. From $\sum_{i=1}^{N}x_{i}\partial_{i}L\left(x\right)=0$ it follows that $\sum_{i=1}^{N}x_{i}\partial_{i}$ commutes with $L\left(x\right)$ and together with $\gamma=S_{1}\left(\tau\right)/N$ completes the proof. to get the stated formula.

Lemma 8

For $1\leq i\leq N$

[TABLE]

Proof. In the square of formula (9) group the terms as

[TABLE]

because

[TABLE]

and $M\left(w,x\right)$ is locally constant in $x$. Next consider

[TABLE]

because $\sigma^{M}\left(i,j\right)\dfrac{x_{i}}{x_{i}-x_{j}}=\dfrac{x_{j}}{x_{j}-x_{i}}\sigma^{M}\left(i,j\right)$ and $\sigma^{M}\left(i,j\right)^{2}=I$ (by Proposition 5).

More detailed analysis of the terms in line (14) shows that there are four different coefficients of $\dfrac{\kappa^{2}}{\left(x_{i}-x_{j}\right)\left(x_{j}-x_{k}\right)}\sigma^{M}\left(\left(i,j\right)\left(i,k\right)\right)$ depending on the numerical order of $i,j,k$:

$x_{i}x_{j}$ if $j<k<i$ or $k<j<i$, 2. 2.

$x_{i}x_{k}$ if $j<i<k$, 3. 3.

$x_{j}x_{k}$ if $i<j<k$ or $i<k<j$, 4. 4.

$x_{j}^{2}$ if $k<i<j$.

The next step is to sum over $1\leq i\leq N$. Lines (12,13) sum to zero by using Formula (10). We show that all the terms in line (14) sum to zero. This is a sum over all cycles of order $3$. Any $3$-cycle is of the form $\left(a,b,c\right)$ with $1\leq a<b<c\leq N$ or $1\leq a<c<b\leq N$. Each $3$-cycle appears three times in the sum since

[TABLE]

If $a<b<c$ then by the above formulae the coefficient of $\kappa^{2}\sigma^{M}\left(\left(a,b,c\right)\right)$ is

[TABLE]

If $a<c<b$ then the coefficient of $\kappa^{2}\sigma^{M}\left(\left(a,b,c\right)\right)$ is

[TABLE]

Each pair $\left\{i,j\right\}$ appears twice in the sum of the terms in (11). We have proven the following:

Theorem 9

The $L\left(x\right)$ conjugate of $\sum_{i=1}^{N}\left(\mathcal{U}_{i}-1-\kappa\gamma\right)^{2}$ is

[TABLE]

Thus $\mathcal{H}_{M}$ agrees with $\mathcal{H}$ when applied to $L\left(x\right)p\left(x\right)$ where $p$ is a symmetric ($\tau\left(w\right)p\left(xw\right)=p\left(x\right))$ polynomial. By Proposition 3 the operators $L\left(x\right)\sum_{i=1}^{N}\mathcal{U}_{i}^{m}L\left(x\right)^{-1}$ commute with $\mathcal{H}_{M}$ and with $\sigma^{M}\left(w\right)$ for $w\in\mathcal{S}_{N}$, for $m=1,2,3,\ldots$ and $L\left(x\right)p\left(x\right)$ is an eigenfunction of $\mathcal{H}_{M}$ for any NSJP $p\left(x\right)$.

4 Hermitian forms and nonsymmetric Jack polynomials

The results in this section come from [4],[5],[6]. To obtain square-integrable and mutually orthogonal wavefunctions we start with a Hermitian form $\left\langle\cdot,\cdot\right\rangle_{\mathbb{T}}$ for $\mathcal{P}_{\tau}$ with the properties ($f,g\in\mathcal{P}_{\tau};1\leq i\leq N;c\in\mathbb{C};T,T^{\prime}\in\mathcal{Y}\left(\tau\right)$)

[TABLE]

The properties define the form uniquely and imply $\left\langle\mathcal{U}_{i}f,g\right\rangle_{\mathbb{T}}=\left\langle f,\mathcal{U}_{i}g\right\rangle_{\mathbb{T}}$ and thus the orthogonality of the NSJP’s (Theorem 13 below). The form is not defined for all $\kappa$ and need not be positive-definite. The key results from [4] (recall the maximum hook-length $h_{\tau}$ from (1)) are:

Theorem 10

Suppose $-1/h_{\tau}<\kappa<1/h_{\tau}$ and $L_{0}\left(x\right)$ is the solution of (5) satisfying $L_{0}\left(x_{0}\right)=I$ and extended to $\mathbb{T}_{reg}^{N}$ by (6) then there exists a unique positive-definite matrix $B$ such that $B\tau\left(w_{0}\right)=\tau\left(w_{0}\right)B$ and

[TABLE]

Each $f\in\mathcal{P}_{\tau}$ has the expansion $\sum_{T\in\mathcal{Y}\left(\tau\right)}\left\langle T,T\right\rangle^{-1/2}f_{T}\left(x\right)\otimes T$ with $f_{T}\in\mathcal{P}$ and $f\left(x\right)$ is considered as a column vector $\left[f_{T}\right]_{T\in\mathcal{Y}\left(\tau\right)}$ in the integral formula. It is implicit in the theorem that $L_{0}\left(x\right)^{\ast}BL_{0}\left(x\right)$ is integrable, and $B$ depends on $\kappa$. Henceforth we use $\left\|p\right\|^{2}:=\left\langle p,p\right\rangle_{\mathbb{T}}$ (which need not be positive for $\kappa$ outside the above interval).

There is a unique positive-definite matrix $C$ such that $C^{2}=B$; as a consequence $C$ commutes with $\tau\left(w_{0}\right)$ (because there is real polynomial $r\left(t\right)$ such that $r\left(B\right)=C$). We apply the results of the previous section to

[TABLE]

and the integral formula becomes

[TABLE]

Here is an outline of the structure and properties of NSJP’s: The operators $\mathcal{U}_{i}$ have a triangularity property with respect to a partial order on $\mathbb{N}_{0}^{N}$. For $\alpha\in\mathbb{N}_{0}^{N}$ let $\alpha^{+}$ denote the nonincreasing rearrangement of $\alpha$ so that $\alpha^{+}$ is a partition.

Definition 11

The dominance order $\prec$ and the derived order $\vartriangleleft$ on $\mathbb{N}_{0}^{N}$ are given by (i) $\alpha\prec\beta$ if and only if $\sum_{j=1}^{i}\alpha_{j}\leq\sum_{j=1}^{i}\beta_{j},$ for$~{}1\leq i\leq N$ and$~{}\alpha\neq\beta$; (ii) $\alpha\vartriangleleft\beta$ if and only if $\left|\alpha\right|=\left|\beta\right|$, $\alpha^{+}\prec\beta^{+}$ ,or $\alpha^{+}=\beta^{+}$ and $\alpha\prec\beta$.

For example: $\left(3,1,1\right)\vartriangleleft\left(0,2,4\right)\vartriangleleft$ $\left(4,0,2\right)$; while $\left(4,1,1\right),\left(3,3,0\right)$ are not $\vartriangleleft$-comparable.

The NSJP’s are labeled by pairs $\left(\alpha,T\right)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}\left(\tau\right)$ but the leading term involves a twist.

Definition 12

For $\alpha\in\mathbb{N}_{0}^{N}$ the rank function on $\left\{1,\ldots,N\right\}$ is given by

[TABLE]

then $r_{\alpha}\in\mathcal{S}_{N}$ and $r_{\alpha}\alpha=\alpha^{+}$ the nonincreasing rearrangement of $\alpha.$

For example if $\alpha=\left(1,2,1,4\right)$ then $r_{\alpha}=\left[3,2,4,1\right]$ and $r_{\alpha}\alpha=\alpha^{+}=\left(4,2,1,1\right)$ (recall $w\alpha_{i}=\alpha_{w^{-1}\left(i\right)}$ ).

Theorem 13

For $\left(\alpha,T\right)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}\left(\tau\right)$ and for all $\kappa$ except for a discrete subset of $\mathbb{Q}$ there is a unique simultaneous eigenfunction $\zeta_{\alpha,T}\in\mathcal{P}_{\tau}$ of $\left\{\mathcal{U}_{i}\right\}$, homogeneous of degree $\left|\alpha\right|$, such that

[TABLE]

The $\zeta_{\alpha,T}$ are called nonsymmetric Jack polynomials. The condition on $\kappa$ for existence is satisfied if each pair $\left(\alpha,T\right)$ is determined by its spectral vector $\xi_{\alpha,T}:=\left[\alpha_{i}+1+\kappa c\left(r_{\alpha}\left(i\right),T\right)\right]_{i=1}^{N}$; this includes the interval $-1/h_{\tau}\leq\kappa\leq 1/h_{\tau}$. There is an algorithmic approach to the construction based on the Yang-Baxter (directed) graph. The edges involve the adjacent transpositions $s_{i}$, which act by transposition on the spectral vector, and a degree-raising operation which shifts and increments the spectral vector. The nodes of the graph are of the form

[TABLE]

(abbreviated to $\left(\alpha,T\right)$) the root is $\left(0^{N},T_{0},\left[1+\kappa c\left(i,T_{0}\right)\right]_{i=1}^{N},I,1\otimes T_{0}\right)$ where $T_{0}$ is formed by entering $N,N-1,\ldots,1$ column-by-column in the Ferrers diagram. The degree-raising edge uses the map $\Phi:\left(c_{1},c_{2},\ldots,c_{N}\right)\rightarrow\left(c_{2},c_{3},\ldots,c_{N},c_{1}+1\right)$ on $N$-tuples. It is called an* affine step *and is defined by

[TABLE]

[TABLE]

(recall $w_{0}=\left(1,2,\ldots,N\right)$, an $N$-cycle) the leading term is $x^{\Phi\alpha}\otimes\tau\left(w_{0}^{-1}r_{\alpha}^{-1}\right)T$ and $w_{0}^{-1}r_{\alpha}^{-1}=r_{\Phi\alpha}^{-1}$ because $r_{\Phi\alpha}=r_{\alpha}w_{0}$ for any $\alpha$: $r_{\alpha}w_{0}\left(i\right)=r_{\alpha}\left(w_{0}\left(i\right)\right)=r_{\alpha}\left(i+1\right)$ for $1\leq i<N$, $r_{\alpha}w_{0}\left(N\right)=r_{\alpha}\left(1\right)$. For example: $\alpha=\left(0,3,5,0\right)$, $r_{\alpha}=\left[3,2,1,4\right]$; $\Phi\alpha=\left(3,5,0,1\right)$, $r_{\Phi\alpha}=\left[2,1,4,3\right]$.

The other edges are called steps or jumps, both labeled by $s_{i}$: the formulae for both rely on the commutation (3) and the coefficient $b$ is determined by the condition that $s_{i}\zeta_{\alpha,T}-b\zeta_{\alpha,T}$ is an eigenfunction of $\mathcal{U}_{i}$.

If $\alpha_{i}<\alpha_{i+1}$, then the step $s_{i}$ is

[TABLE]

[TABLE]

If $\alpha_{i}=\alpha_{i+1}$, set $j=r_{\alpha}\left(i\right)$, so that $j+1=r_{\alpha}\left(i+1\right)$ and $s_{i}r_{\alpha}^{-1}=r_{\alpha}^{-1}s_{j}$. Thus $\xi_{\alpha,T}\left(i\right)=\alpha_{i}+1+\kappa c\left(j,T\right)$ and $\xi_{\alpha,T}\left(i+1\right)=\alpha_{i}+1+\kappa c\left(j+1,T\right)$. Set

[TABLE]

If $b^{\prime}=1$ {$\mathrm{rw}\left(j,T\right)=\mathrm{rw}\left(j+1,T\right)$} or $-1$ {$\mathrm{cm}\left(j,T\right)=\mathrm{cm}\left(j+1,T\right)$} then $s_{i}\zeta_{\alpha,T}=\zeta_{\alpha,T}$ or $-\zeta_{\alpha,T}$ respectively. Otherwise let $T^{\left(j\right)}$ denote the result of interchanging $j$ and $j+1$ in $T$. If $0<b^{\prime}\leq\frac{1}{2}$, that is, $\mathrm{rw}\left(j,T\right)<\mathrm{rw}\left(j+1,T\right)$ (and $\mathrm{cm}\left(j,T\right)>\mathrm{cm}\left(j+1,T\right)$; if one takes the $\left(1,1\right)$ cell of $T$ as northwest then $j$ is northeast of $j+1$) then the jump $s_{i}$ is (“jump” suggests jumping from one tableau to another)

[TABLE]

[TABLE]

The leading term is transformed $s_{i}\left(x^{\alpha}\otimes\tau\left(r_{\alpha}^{-1}\right)T\right)=\left(xs_{i}\right)^{\alpha}\otimes\tau\left(s_{i}r_{\alpha}^{-1}\right)T=x^{\alpha}\otimes\tau\left(r_{\alpha}^{-1}\right)\tau\left(s_{j}\right)T$ and $\tau\left(s_{j}\right)T=T^{\left(j\right)}+b^{\prime}T$. The jump applies to the situation $\alpha=0^{N}$ and provides the transformation formulae for $s_{i}$ acting on $T\in\mathcal{Y}\left(\tau\right)$ (that is, on $1\otimes T$ and $r_{\boldsymbol{\alpha}}\left(i\right)=i$).

Example 14

Let $N=3,\tau=\left(2,1\right)$ and T_{0}=\begin{array}[c]{cc}3&1\\ 2&\end{array},T_{1}=\begin{array}[c]{cc}3&2\\ 1&\end{array}, and consider $\left(\alpha,T\right)=\left(\left(0,1,1\right),T_{0}\right).$ Then $r_{\alpha}=\left[3,1,2\right]$ and $\xi_{\alpha,T_{0}}=\left[1,2+\kappa,2-\kappa\right]$. The step $s_{1}$ is $\zeta_{\left(1,0,1\right),T_{0}}=s_{1}\zeta_{\alpha,T_{0}}+\dfrac{\kappa}{1+\kappa}\zeta_{\alpha,T_{0}}$; for the jump $s_{2}$ one finds $j=1,b^{\prime}=\frac{1}{2}$ and $\zeta_{\left(0,1,1\right),T_{1}}=s_{2}\zeta_{\alpha,T_{0}}-\frac{1}{2}\zeta_{\alpha,T_{0}}$. Note $\xi_{\left(0,1,1\right),T_{1}}=\left[1,2-\kappa,2+\kappa\right]$.

The hypotheses on the Hermitian form (15) imply

[TABLE]

and thus $\left\langle\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\right\rangle_{\mathbb{T}}=0$, (for permitted values of $\kappa$). The orthogonality provides an inductive process for computing $\left\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\right\rangle_{\mathbb{T}}$: for the step $s_{i}$ with $\alpha_{i}<\alpha_{i+1}$ we have $s_{i}\zeta_{\alpha,T}=\zeta_{s_{i}\alpha,T}+b\zeta_{\alpha,T}$ (where $b=\frac{\kappa}{\xi_{\alpha,T}\left(i\right)-\xi_{\alpha,T}\left(i+1\right)}$) and

[TABLE]

A similar formula holds for the jump ($\alpha_{i}=\alpha_{i+1}$). For the affine step, the hypotheses (15) imply $\left\|\zeta_{\Phi\alpha,T}\right\|^{2}=\left\|\zeta_{\alpha,T}\right\|^{2}$. Together with $\left\langle 1\otimes T,1\otimes T^{\prime}\right\rangle_{\mathbb{T}}=\left\langle T,T^{\prime}\right\rangle_{0}$ this procedure leads to formulae for all $\left\|\zeta_{\alpha,T}\right\|^{2}$.

Theorem 15

For $\lambda\in\mathbb{N}_{0}^{N,+}$ ($\lambda_{1}\geq\lambda_{2}\ldots\geq\lambda_{N}$) and $T\in\mathcal{Y}\left(\tau\right)$

[TABLE]

There is an additional factor for nonpartition indices.

Definition 16

For $\alpha\in\mathbb{N}_{0}^{N},T\in\mathcal{Y}\left(\tau\right)$ and $\varepsilon=\pm 1$ set

[TABLE]

Theorem 17

Suppose $\alpha\in\mathbb{N}_{0}^{N},T\in\mathcal{Y}\left(\tau\right)$ then $\left\|\zeta_{\alpha,T}\right\|^{2}=\left(\mathcal{E}_{1}\left(\alpha,T\right)\mathcal{E}_{-1}\left(\alpha,T\right)\right)^{-1}\left\|\zeta_{\alpha^{+},T}\right\|^{2}$.

It is important that $-1/h_{\tau}<\kappa<1/h_{\tau}$ implies $\left\|\zeta_{\alpha,T}\right\|^{2}>0$ for all $\left(\alpha,T\right)$ and thus $\left\langle\cdot,\cdot\right\rangle_{\mathbb{T}}$ is positive-definite. Observe that the value of $\left\|\zeta_{\alpha,T}\right\|^{2}$ depends only on the differences $\alpha_{i}-\alpha_{j}$. This is a consequence of the torus property $\left\langle x_{i}f,x_{i}g\right\rangle_{\mathbb{T}}=\left\langle f,g\right\rangle_{\mathbb{T}}$ and the commutation (where $e_{N}:=x_{1}x_{2}\cdots x_{N}$ )

[TABLE]

Thus $\mathcal{U}_{i}\left(e_{N}^{m}\zeta_{\alpha,T}\right)=\left(m+\alpha_{i}+\kappa c\left(r_{\alpha}\left(i\right),T\right)\right)e_{N}^{m}\zeta_{\alpha,T}$, and $e_{N}^{m}\zeta_{\alpha,T}$ is a simultaneous eigenfunction of $\left\{\mathcal{U}_{i}\right\}$ with the same eigenvalues and the same leading term as $\zeta_{\alpha+m\boldsymbol{1},T}$ for $m\geq 0$ (with $\boldsymbol{1}:=\left(1,1,\ldots,1\right)\in\mathbb{N}_{0}^{N}$). Hence $\zeta_{\alpha+m\boldsymbol{1},T}=e_{N}^{m}\zeta_{\alpha,T}$. There are Laurent polynomial eigenfunctions of $\left\{\mathcal{U}_{i}\right\}$. The structure of NSJP’s is extended to $V_{\tau}$-valued Laurent polynomials, thereby producing a basis:

Definition 18

Suppose $\alpha\in\mathbb{Z}^{N}$ then set $\zeta_{\alpha,T}=e_{N}^{-m}\zeta_{\alpha+m\boldsymbol{1},T}$ where $m\in\mathbb{N}_{0}$ and satisfies $m\geq-\min_{j}\alpha_{i}$. This is well-defined since $\alpha+m\boldsymbol{1}\in\mathbb{N}_{0}^{N}$ and $\zeta_{\alpha+k\boldsymbol{1},T}=e_{N}^{k}\zeta_{\alpha,T}$ for $k\in\mathbb{N}_{0}$.

5 Symmetric vector-valued polynomials

For an arbitrary $\left(\alpha,T\right)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}\left(\tau\right)$ we can define a symmetric polynomial simply by averaging: $p=\frac{1}{N!}\sum_{w\in\mathcal{S}_{N}}w\zeta_{\alpha,T}$. From Proposition 3 it follows that $p$ is an eigenfunction of $\sum_{i=1}^{N}\mathcal{U}_{i}^{m}$ for each $m=1,2,3,\ldots$. The idea of using this method to construct Jack polynomials from the scalar nonsymmetric Jack polynomials is due to Baker and Forrester [2]; the usual Jack parameter is $\alpha=1/\kappa$. It is possible that for some $\left(\alpha,T\right)$ the sum $p=0$, and for some pairs $\left(\alpha,T\right)$ and $\left(\beta,T^{\prime}\right)$ that the sums agree up to multiplication by a constant. In this section we present the structure of Jack polynomials, the assignment of unique labels, and orthogonality properties (henceforth, unmodified “Jack” implies symmetry). Multiplication by $L\left(x\right)$ will yield symmetric eigenfunctions of $\mathcal{H}$, the vector-valued wavefunctions. The results are mostly from [5, Sec. 5.2]. The Jack polynomials correspond to certain connected components of the Yang-Baxter graph after the affine jumps are removed.

Definition 19

For $\alpha\in\mathbb{N}_{0}^{N},T\in\mathcal{Y}\left(\tau\right)$ define $\left\lfloor\alpha,T\right\rfloor$ to be the filling of the Ferrers diagram of $\tau$ obtained by replacing $i$ by $\alpha_{i}^{+}$ in $T$, for all $i$.

Obviously $\left\lfloor\alpha,T\right\rfloor=\left\lfloor\alpha^{+},T\right\rfloor$.

Example 20

Let $\tau=\left(3,2\right),\alpha=\left(1,4,2,0,3\right)$ and

[TABLE]

Proposition 21

([5, Prop. 5.2]) $\left(\alpha,T\right)$ and $\left(\beta,T^{\prime}\right)$ are connected by edges and jumps (without regard to the orientation) if and only if $\left\lfloor\alpha,T\right\rfloor=\left\lfloor\beta,T^{\prime}\right\rfloor$.

From the properties of steps and jumps it follows that the spectral vectors of $\left(\alpha,T\right)$ and $\left(\beta,T^{\prime}\right)$ are permutations of each other. Set $\mathcal{T}\left(\alpha,T\right)=\left\{\left(\beta,T^{\prime}\right):\left\lfloor\beta,T^{\prime}\right\rfloor=\left\lfloor\alpha,T\right\rfloor\right\}$, the set of nodes in the connected component.

A tableau is column-strict if the entries are increasing in each column, and nondecreasing in each row.

Theorem 22

For $\left(\alpha,T\right)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}\left(\tau\right)$ the $\mathrm{span}\left\{\zeta_{\beta,T^{\prime}}:\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\alpha,T\right)\right\}$ contains a unique nonzero symmetric polynomial if and only if $\left\lfloor\alpha,T\right\rfloor$ is column-strict.

As usual in this context, unique means up to multiplication by a scalar. Suppose $\lambda\in\mathbb{N}_{0}^{N,+}$ and consider the sum $p=\sum\limits_{\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\lambda,T\right)}a\left(\beta,T^{\prime}\right)\zeta_{\beta,T^{\prime}}$ subject to the conditions $s_{i}p=p$ for $1\leq i<N$ (sufficing for symmetry).

Suppose there is a step or jump $s_{i}$ from $\left(\beta,T^{\prime}\right)$ to $\left(\gamma,T^{\prime\prime}\right)$ then $\zeta_{\left(\gamma,T^{\prime\prime}\right)}=s_{i}\zeta_{\left(\beta,T^{\prime}\right)}-b\zeta_{\left(\beta,T^{\prime}\right)}$ for some $b$; this implies $s_{i}\zeta_{\left(\gamma,T^{\prime\prime}\right)}=-b\zeta_{\left(\beta,T^{\prime}\right)}+\left(1-b^{2}\right)\zeta_{\left(\gamma,T^{\prime\prime}\right)}$. The condition $s_{i}\left(a\left(\beta,T^{\prime}\right)\zeta_{\beta,T^{\prime}}+a\left(\gamma,T^{\prime\prime}\right)\zeta_{\gamma,T^{\prime\prime}}\right)=a\left(\beta,T^{\prime}\right)\zeta_{\beta,T^{\prime}}+a\left(\gamma,T^{\prime\prime}\right)\zeta_{\gamma,T^{\prime\prime}}$ implies $a\left(\beta,T^{\prime}\right)=\left(1+b\right)a\left(\gamma,T^{\prime\prime}\right)$. The column strictness hypothesis implies that $b=-1$ can not occur. The relation is used in an inductive evaluation of $a\left(\beta,T^{\prime}\right)$ once the beginning and end have been identified.

Definition 23

For $\alpha\in\mathbb{N}^{N}$ and $T\in Y\left(\tau\right)$ let

[TABLE]

Thus a step reduces $\mathrm{inv}\left(\alpha\right)$ by $1$ (that is, $\mathrm{inv}\left(s_{i}\alpha\right)=\mathrm{inv}\left(\alpha\right)-1$) and a jump reduces $\mathrm{inv}\left(T\right)$ by $1$ (in Example 14 $\mathrm{inv}\left(T_{0}\right)=1$ and $\mathrm{inv}\left(T_{1}\right)=0$). Hence the root $\left(\alpha,T\right)$ of $\in\mathcal{T}\left(\alpha,T\right)$ has maximum $\mathrm{inv}\left(\alpha\right)+\mathrm{inv}\left(T\right)$ and the sink has minimum $\mathrm{inv}\left(\alpha\right)+\mathrm{inv}\left(T\right)$. Clearly $\mathrm{inv}\left(\alpha\right)$ is minimized at $\alpha^{+}$ and maximized at $\alpha^{-}$, the nondecreasing rearrangement of $\alpha$. In [5, Def. 5.6] it is shown that there are unique tableaux $T_{R},T_{S}$ in $\in\mathcal{T}\left(\alpha,T\right)$ such that $\left(\alpha^{-},T_{R}\right)$ is the root and $\left(\alpha^{+},T_{S}\right)$ is the sink (maximizes, respectively minimizes $\mathrm{inv}\left(\beta\right)+\mathrm{inv}\left(T^{\prime}\right)$ for $\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\alpha,T\right)$). The formulae for $T_{R}$ and $T_{S}$ are (with $\mathcal{T}=\left\lfloor\alpha,T\right\rfloor$)

[TABLE]

[TABLE]

Example 24

$\alpha=\left(3^{3},2^{3},1,0\right),$

[TABLE]

As motivation for the formulae for $a\left(\beta,T^{\prime}\right)$ in the sum suppose $\beta_{i}<\beta_{i+1}$ (and $\varepsilon=\pm 1$) then

[TABLE]

where $\zeta_{s_{i}\beta,T}=s_{i}\zeta_{\beta,T}-b\zeta_{\beta,T}$. We introduce two functions on $\mathcal{Y}\left(\tau\right)$ to deal analogously with jumps:

Definition 25

For $T\in\mathcal{Y}\left(\tau\right)$ and $\varepsilon=\pm 1$ set

[TABLE]

From (2) $\left\langle T,T\right\rangle_{0}=\mathcal{C}_{1}\left(T\right)\mathcal{C}_{-1}\left(T\right)$. In the jump with $\beta_{i}=\beta_{i+1},r_{\beta}\left(i\right)=j$ and $c\left(j,T\right)-c\left(j+1,T\right)\geq 2$ (as in 17) $T^{\left(j\right)}$ has $j$ and $j+1$ interchanged so that $c\left(j,T^{\left(j\right)}\right)-c\left(j+1,T^{\left(j\right)}\right)=c\left(j+1,T\right)-c\left(j,T\right)\leq-2.$ Then

[TABLE]

so that $\mathcal{C}_{-1}\left(T^{\left(j\right)}\right)=\mathcal{C}_{-1}\left(T\right)\left(1+b\right)$ where $b=\left(c\left(j,T\right)-c\left(j+1,T\right)\right)^{-1}$. From these relations it can be shown:

Proposition 26

Suppose $\left(\alpha,T\right)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}\left(\tau\right)$ and $\left\lfloor\alpha,T\right\rfloor$ is column-strict then

[TABLE]

is symmetric and nonzero.

To proceed with the analysis we impose a normalization and then find a closed formula for the squared-norm $\left\|\cdot\right\|^{2}$. Replace $\alpha$ by $\lambda=\alpha^{+}$ and use the sink $\left(\lambda,T_{S}\right)$ as normalization by requiring that the coefficient of $x^{\lambda}v_{T_{S}}$ is $1$. From the sink property and the $\vartriangleright$-triangularity of the NSJP’s it follows that $x^{\lambda}v_{T_{S}}$ appears only in $\zeta_{\lambda,T_{S}}$ in the sum $p$, with coefficient $1$. Thus define (for $\lambda\in\mathbb{N}_{0}^{N,+}$)

[TABLE]

By orthogonality $\left\|J_{\lambda,T_{S}}\right\|^{2}=\sum\limits_{\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\lambda,T_{S}\right)}\left(\frac{\mathcal{C}_{-1}\left(T_{S}\right)}{\mathcal{C}_{-1}\left(T^{\prime}\right)}\mathcal{E}_{-1}\left(\beta,T^{\prime}\right)\right)^{2}\left\|\zeta_{\beta,T^{\prime}}\right\|^{2}$; fortunately there is a formula without summation. Suppose $\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\lambda,T_{S}\right)$ and $J_{\lambda,T_{S}}=c\sum\limits_{w\in\mathcal{S}_{N}}w\zeta_{\beta,T^{\prime}}$ then

[TABLE]

if we write $J_{\lambda,T_{S}}=\sum\limits_{\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\lambda,T_{S}\right)}a\left(\beta,T^{\prime}\right)\zeta_{\beta,T^{\prime}}$ then $\left\|J_{\lambda,T_{S}}\right\|^{2}=\left(N!c\right)a\left(\beta,T^{\prime}\right)\left\|\zeta_{\beta,T^{\prime}}\right\|^{2}$ and $c$ can be determined by careful choice of $\left(\beta,T^{\prime}\right)$. Consider the stabilizer group $G_{\lambda,T_{S}}$ of $\zeta_{\lambda,T_{S}}$ ($w\in G_{\lambda,T_{S}}$ implies $w\zeta_{\lambda,T_{S}}=\zeta_{\lambda,T_{S}}$). The group is generated by $\left\{s_{i}:\lambda_{i}=\lambda_{i+1},\mathrm{rw}\left(i,T_{S}\right)=\mathrm{rw}\left(i+1,T_{S}\right)\right\}$. It was shown ([5, Prop. 5.11]) that the coefficient of $\zeta_{\lambda,T_{S}}$ in $\sum\limits_{w\in\mathcal{S}_{N}}w\zeta_{\lambda^{-},T_{R}}$ is $\#G_{\lambda,T_{S}}$, hence that $\sum\limits_{w\in\mathcal{S}_{N}}w\zeta_{\lambda^{-},T_{R}}=\#G_{\lambda,T_{S}}J_{\lambda,T_{S}}$. We deduce

[TABLE]

Also

[TABLE]

and $\dfrac{\left\|\zeta_{\lambda,T_{R}}\right\|^{2}}{\left\langle T_{R},T_{R}\right\rangle_{0}}=\dfrac{\left\|\zeta_{\lambda,T_{S}}\right\|^{2}}{\left\langle T_{S},T_{S}\right\rangle_{0}}$ (from Theorem 15). From (2) it follows that $\dfrac{\left\langle T_{R},T_{R}\right\rangle_{0}}{\left\langle T_{S},T_{S}\right\rangle_{0}}=\dfrac{\mathcal{C}_{1}\left(T_{R}\right)\mathcal{C}_{-1}\left(T_{R}\right)}{\mathcal{C}_{1}\left(T_{S}\right)\mathcal{C}_{-1}\left(T_{S}\right)}$, and $\dfrac{N!}{\#G_{\lambda,T_{S}}}=\#\mathcal{T}_{\lambda,T_{S}}$. To summarize:

Theorem 27

Suppose $\left(\lambda,T\right)\in\mathbb{N}_{0}^{N,+}\times\mathcal{Y}\left(\tau\right)$ and $\left\lfloor\lambda,T\right\rfloor$ is column-strict. Define $T_{R}$ and $T_{S}$ by formulae (18) and (19) then

[TABLE]

Suppose $\left\lfloor\lambda,T_{S}\right\rfloor,\left\lfloor\lambda^{\prime},T_{S}\right\rfloor$ are unequal column-strict tableaux. By definition $\mathcal{T}\left(\lambda,T_{S}\right)\cap\mathcal{T}\left(\lambda^{\prime},T_{S}\right)=\varnothing$ and from the mutual orthogonality of the terms in the sums (20) it follows that $\left\langle J_{\lambda,T_{S}},J_{\lambda^{\prime},T_{S}}\right\rangle_{\mathbb{T}}=0$.

Multiplication of $J_{\lambda,T_{S}}$ by $e_{N}^{m}$ produces the Jack polynomial $J_{\lambda+m\boldsymbol{1},T_{S}}$ (see Definition 18); here $m=-1,-2,\ldots$is valid and defines Jack Laurent polynomials. The expression $e_{N}^{m}J_{\lambda,T_{S}}$ is made unique by the requirement $\lambda_{N}=0$.

We see that there is a unique symmetric polynomial $p_{\lambda,T}$ of minimum degree: the tableau $\left\lfloor\lambda,T\right\rfloor$ has the entry $i-1$ in each box in row $\#i$. The degree is $n\left(\tau\right)=\sum_{i\geq 1}\left(i-1\right)\tau_{i}$.

Remark 28

A weak reverse tableau of shape $\tau$ and weight $n$ has its entries nondecreasing in each row and in each column and the sum of the entries is $n$. Such a tableau can be transformed to a column-strict tableau by adding $i-1$ to each entry in row $\#i$ for $1\leq i\leq\ell\left(\tau\right).$ The number of the weak reverse tableaux is the coefficient of $z^{n}$ in $H_{\tau}\left(z\right):=\prod_{\left(i,j\right)\in\tau}\left(1-z^{h\left(i,j\right)}\right)^{-1}$ (see [9, p.379], $h\left(i,j\right)$ from Formula (1)). Thus the number of Jack polynomials of degree $n$ is the coefficient of $z^{n}$ in $z^{n\left(\tau\right)}H_{\tau}\left(z\right)$. For example take $\tau=\left(3,2\right)$ then $z^{n\left(\tau\right)}H_{\tau}\left(z\right)=z^{2}\left\{\left(1-z\right)^{2}\left(1-z^{2}\right)\left(1-z^{3}\right)\left(1-z^{4}\right)\right\}^{-1}$. The analogous number when $\lambda_{N}=0$ is the coefficient of $z^{n}$ in $\left(1-z^{N}\right)z^{n\left(\tau\right)}H_{\tau}\left(z\right)$.

If $\left(\beta,T^{\prime}\right)\in\mathcal{T}\left(\lambda,T_{S}\right)$ then the spectral vector $\xi_{\beta,T^{\prime}}$ is a permutation of $\xi_{\lambda,T_{S}}$ thus $\sum_{i=1}^{N}\mathcal{U}_{i}^{m}J_{\lambda,T_{S}}=\sum_{j=1}^{N}\xi_{j}\left(\lambda,T_{S}\right)^{m}J_{\lambda,T_{S}}$ for $m=1,2,3,\ldots$. and

[TABLE]

Set $S_{2}:=\sum_{i=1}^{\ell\left(\tau\right)}c\left(i,T_{S}\right)^{2}=\frac{1}{6}\sum_{i=1}^{\ell\left(\tau\right)}\tau_{i}\left\{\left(\tau_{i}-1\right)\left(\tau_{i}-2\right)-6\left(\tau_{i}-i\right)\left(i-1\right)\right\}$. The eigenvalue can be written as $\sum_{i=1}^{N}\lambda_{i}^{2}+2\kappa\sum_{i=1}^{N}\lambda_{i}\left(c\left(i,T_{S}\right)-\gamma\right)+\kappa^{2}\left(S_{2}-N\gamma^{2}\right).$ In the trivial case $\tau=\left(N\right)$ the last term becomes $\frac{1}{12}\kappa^{2}N\left(N^{2}-1\right)$. The effect of multiplying by $e_{N}^{m}$ on the eigenvalue is

[TABLE]

This is minimized over $m$ when $m$ is the nearest integer to $-\sum_{i=1}^{N}\lambda_{i}/N$.

6 Symmetric wavefunctions

In the notation of Section 5 there is a set of mutually orthogonal wavefunctions $L\left(x\right)J_{\lambda,T_{S}}\left(x\right)$ such that

[TABLE]

by Theorem 9. Thus

[TABLE]

is a probability density function on $\mathbb{T}^{N}$. Since multiplication of $L\left(x\right)J_{\lambda,T_{S}}\left(x\right)$ by powers of $e_{N}$ does not change the density function, we can assume $\lambda_{N}=0$. In contrast to the scalar case the matrix $L\left(x\right)$ has singularities of order $\left|x_{i}-x_{j}\right|^{\pm\kappa}$ in neighborhoods of points $x$ with $x_{i}=x_{j}$ and all other $x_{k}$ being pairwise distinct. Nevertheless we can show that the symmetric wavefunctions are bounded in such sets when $0<\kappa<\frac{1}{h_{\tau}}$. By the invariance it suffices to prove this near $\left\{x:x_{N-1}=x_{N}\right\}$ in the fundamental chamber. More precisely let $\delta>0$ and define

[TABLE]

Theorem 29

Suppose $0<\kappa<\frac{1}{h_{\tau}}$ and $\left(\lambda,T_{S}\right)$ is as in Theorem 27 then $L\left(x\right)J_{\lambda,T_{S}}\left(x\right)$ is uniformly bounded in $\Omega_{\delta}$.

The proof depends on a power series formulation for $L$ proven in [4, Sec. 5]. Arrange $\mathcal{Y}\left(\tau\right)$ linearly listing the tableaux $T$ with $c\left(N-1,T\right)=-1$ first (that is, $\mathrm{cm}\left(N-1,T\right)=1,\mathrm{rw}\left(N-1,T\right)=2$). This results in the matrix representation of $\tau\left(N-1,N\right)$ being

[TABLE]

where $n_{\tau}:=\dim V_{\tau}=\#\mathcal{Y}\left(\tau\right)$ and $m_{\tau}$ is given by $\mathrm{tr}\left(\tau\left(N-1,N\right)\right)=n_{\tau}-2m_{\tau}$. From the sum $\sum_{i<j}\tau\left(i,j\right)=S_{1}\left(\tau\right)I$ it follows that $\binom{N}{2}\mathrm{tr}\left(\tau\left(N-1,N\right)\right)=S_{1}\left(\tau\right)n_{\tau}$ and $m_{\tau}=n_{\tau}\left(\frac{1}{2}-\frac{S_{1}\left(\tau\right)}{N\left(N-1\right)}\right)$. (The traces of the transpositions are equal because they are conjugate to each other.) The property of a matrix commuting or anti-commuting with $\sigma:=\tau\left(N-1,N\right)$ is used in the argument; to express this neatly we introduce the $\sigma$-block decomposition $\left(m_{\tau}+\left(n_{\tau}-m_{\tau}\right)\right)\times\left(m_{\tau}+\left(n_{\tau}-m_{\tau}\right)\right)$ of a matrix

[TABLE]

Then $\sigma\alpha\sigma=\alpha$ if and only if $\alpha_{12}=O=\alpha_{21}$ and $\sigma\alpha\sigma=-\alpha$ if and only if $\alpha_{11}=O=\alpha_{22}$. For $z_{1},z_{2}\in\mathbb{C}$ let

[TABLE]

We showed in [4, Sec. 5] that there exist matrix coefficients $\alpha_{n}\left(x^{\prime}\right)$ with $x^{\prime}:=\left(x_{1},\ldots,x_{N-2},\frac{x_{N-1}+x_{N}}{2}\right)$, analytic on the closure of $\Omega_{\delta}\cup\Omega_{\delta}\left(N-1,N\right)$, such that (with $z:=\frac{x_{N}-x_{N-1}}{2}$)

[TABLE]

and $\sigma\alpha_{n}\left(x^{\prime}\right)\sigma=\left(-1\right)^{n}\alpha_{n}\left(x^{\prime}\right).$ In particular the $\sigma$-block decomposition of $\alpha_{0}\left(x^{\prime}\right)$ is \left[\begin{array}[c]{cc}\alpha_{0,11}\left(x^{\prime}\right)&O\\ O&\alpha_{0,22}\left(x^{\prime}\right)\end{array}\right]. The series converges absolutely for $\frac{1}{2}\left|x_{N}-x_{N-1}\right|<\min\limits_{1\leq j\leq N-2}\left|x_{j}-\frac{x_{N-1}+x_{N}}{2}\right|$.

Suppose $J_{\lambda,T_{S}}$ is nonzero as in (20) then express $J_{\lambda,T_{S}}\left(x\right)\allowbreak=\sum\limits_{T\in\mathcal{Y}\left(\tau\right)}\frac{1}{\left\langle T,T\right\rangle_{0}^{1/2}}p_{T}\left(x\right)\otimes T$ where each $p_{T}\left(x\right)$ is a scalar polynomial. If $c\left(N-1,T\right)=-1$ then $\tau\left(N-1,N\right)T=-T$ and the relation $\left(N-1,N\right)J_{\lambda,T_{S}}=J_{\lambda,T_{S}}$ implies $p_{T}\left(x\left(N-1,N\right)\right)=-p\left(x\right)$ (see Proposition. 1). In the order of the basis chosen above the first $m_{\tau}$ coefficients of $J_{\lambda,T_{S}}$ are divisible by $x_{N}-x_{N-1}$. Write $J_{\lambda,T_{S}}$ as a column vector $\left[p_{1}^{tr},p_{2}^{tr}\right]^{tr}$ with $p_{1}$ consisting of the first $m_{\tau}$ coordinates. Suppose $\kappa>0$ then the dominant part of $\left(L\left(x\right)J_{\lambda,T_{S}}\left(x\right)\right)^{\ast}L\left(x\right)J_{\lambda,T_{S}}\left(x\right)$ is

[TABLE]

(with $z=\frac{x_{N}-x_{N-1}}{2}$) the omitted terms are of order $\left|x_{N}-x_{N-1}\right|^{1-2\left|\kappa\right|}$. Each component of $p_{1}\left(x\right)$ is divisible by $x_{N}-x_{N-1}$ and thus the first part of the expression is of order $\left|x_{N}-x_{N-1}\right|^{2-2\kappa}$ and the second part is of order $\left|x_{N}-x_{N-1}\right|^{2\kappa}$. Hence $\left(L\left(x\right)J_{\lambda,T_{S}}\left(x\right)\right)^{\ast}L\left(x\right)J_{\lambda,T_{S}}\left(x\right)$ is bounded on $\Omega_{\delta}$. It may be suspected that the bound holds for all of $\mathbb{T}^{N}$ but the behaviour of $L\left(x\right)$ near a point with multiple repeated entries (say $x_{N-2}=x_{N-1}=x_{N}$) is complicated and the power series method used here does not apply.

6.1 Minimal degree symmetric polynomials

To produce the minimal degree $J_{\lambda,T_{S}}$, or equivalently, the minimal degree column-strict tableau of shape $\tau,$ the entries in row $\#i$ all equal $i-1$. The corresponding $T_{S}$ has the numbers $N,N-1,\ldots,2,1$ entered row-by-row, and $T_{R}=T_{S}$. With $l=\ell\left(\tau\right)$ and using superscripts to denote multiplicity $\lambda=\left(\left(l-1\right)^{\tau_{l}},\left(l-2\right)^{\tau_{l-1}},\ldots,1^{\tau_{2}},0^{\tau_{1}}\right)$. There is an explicit formula for $J_{\lambda,T_{S}}$. It is derived from Proposition 1 and involves the Specht polynomials. For $1\leq n_{1}\leq n_{2}\leq N$ define the alternating polynomial $a\left(x;n_{1},n_{2}\right)=\prod\limits_{n_{1}\leq i<j\leq n_{2}}\left(x_{i}-x_{j}\right)$ (the empty product $a\left(x;n_{1},n_{1}\right)=1$). Denote the transpose of the partition $\tau$ by $\tau^{\prime}$ then $\tau_{1}^{\prime}=\ell\left(\tau\right)$. Form $p_{T_{0}}$ as the product of the alternating polynomials for each column of $T_{0}$: that is, set $k_{j}=N-\sum_{i=1}^{j}\tau_{j}^{\prime},0\leq j\leq\tau_{1}$ and

[TABLE]

Then the other polynomials $p_{T}$ are produced by the formulae in the Proposition and the minimal $J_{\lambda,T_{S}}=\left\langle T_{S},T_{S}\right\rangle_{0}\sum\limits_{T^{\prime}\in\mathcal{Y}\left(\tau\right)}\frac{1}{\left\langle T^{\prime},T^{\prime}\right\rangle_{0}}p_{T^{\prime}}\left(x\right)\otimes T^{\prime}$ (see [5, Sec. 5.4]). For example let $\tau=\left(2,2\right)$ then

[TABLE]

[TABLE]

because $b=\left(c\left(2,T_{0}\right)-c\left(3,T_{0}\right)\right)^{-1}=\frac{1}{2}$. Observe that $s_{1}p_{T_{1}}=p_{T_{1}}=s_{3}p_{T_{1}}$. Also $\left\langle T_{0},T_{0}\right\rangle_{0}=1$ and $\left\langle T_{1},T_{1}\right\rangle_{0}=\frac{3}{4}$ (see (2), $\lambda=\left(1,1,0,0\right)$ and $J_{\lambda,T_{1}}=\frac{3}{4}p_{T_{0}}\otimes T_{0}+p_{T_{1}}\otimes T_{1}$.

There is a formula for the minimal $\left(\lambda,T_{S}\right)$ proven in [3, Thm. 8]:

[TABLE]

where $\mathrm{leg}\left(i,j\right):=\#\left\{l:l>i,\tau_{l}\geq j\right\}$ (equivalently $\tau_{j}^{\prime}-i$), and $\left(m\right)_{n}$ is the Pochhammer symbol $\prod_{j=1}^{n}\left(m+j-1\right)$.

The energy eigenvalue $\sum_{i=1}^{N}\lambda_{i}^{2}+2\kappa\sum_{i=1}^{N}\lambda_{i}\left(c\left(i,T_{S}\right)-\gamma\right)+\kappa^{2}\left(S_{2}-N\gamma^{2}\right)$ from (21) specializes to

[TABLE]

for the minimal degree $J_{\lambda,T_{S}}$.

6.2 Example

Let $N=4,\tau=\left(2,2\right),$ $\mathcal{Y}\left(\tau\right)=\left\{T_{0},T_{1}\right\}$ see (22), $h_{\tau}=3$. The system (5) can be reduced to a hypergeometric equation in one variable by the substitution $\zeta\left(x\right)=\dfrac{\left(x_{1}-x_{2}\right)\left(x_{3}-x_{4}\right)}{\left(x_{1}-x_{3}\right)\left(x_{2}-x_{4}\right)}$. The bounds $0<\zeta\left(x\right)<1$ hold in the fundamental domain $\mathcal{C}_{0}$. A fundamental solution $L_{F}$ is given in terms of the functions

[TABLE]

[TABLE]

Then the solution $L\left(x\right)$ which satisfies (16) is up to a positive multiplicative constant

[TABLE]

where $\gamma\left(\kappa\right):=\frac{\Gamma\left(1+2\kappa\right)^{2}}{\Gamma\left(1+\kappa\right)\Gamma\left(1+3\kappa\right)}$; observe that $\gamma\left(\kappa\right)>0$ for $\kappa>-\frac{1}{3}$. As yet the problem of determining the normalizing constant is still open. The purpose of the example is to demonstrate the qualitative difference of $L\left(x\right)$ from the scalar case, so the underlying computations are not presented here. The method uses the known transformation properties of the hypergeometric series for the change-of-variable $\zeta\rightarrow 1-\zeta$ , since $\zeta\left(xw_{0}\right)=1-\zeta\left(x\right)$ where $w_{0}=\left(1,2,3,4\right)$, a $4$-cycle.