You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


[**GENERALIZATION OF A RESULT OF MATSUO AND CHEREDNIK TO THE CALOGERO-SUTHERLAND-MOSER INTEGRABLE MODELS WITH EXCHANGE TERMS[^1]**]{}

A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with $\delta$-function interaction.

Introduction
============

The Calogero integrable system [@cal1] consists of $N$ nonrelativistic particles on the line interacting through a two-body potential of the inverse square type, $$V_C^{(k)} = k(k-1) \sum_{\scriptstyle i,j=1 \atop \scriptstyle i\ne j}^N
  \frac{1}{(x_i-x_j)^2}.             \label{eq:1calpot}$$ Together with its trigonometric, hyperbolic, and elliptic generalizations [@cal2]–[@mos], it constitutes the Calogero-Sutherland-Moser (CSM) integrable model, which is related to the root systems of ${\cal A}_{N-1}$ algebras [@pere] and is founding some interesting applications in field-theoretical and condensed-matter contexts.

A breakthrough in the study of CSM systems occurred some three years ago when Polychronakos [@poly] and Brink [*et al*]{} [@brink1] independently introduced an exchange operator formalism, leading to a set of $N$ commuting first-order differential operators, known in the mathematical literature as Dunkl operators [@dunkl]. Such operators led to Hamiltonians with exchange terms, connected with the spin generalizations of the CSM systems [@ber]. Spectra and wave functions of the latter can be obtained by simultaneously diagonalizing the $N$ commuting Dunkl operators.

At approximately the same time, another approach to CSM systems made its appearance. It was based upon the Knizhnik-Zamolodchikov (KZ) equations, previously introduced in conformal field theory [@kz]. The latter can be written as $$\partial_i\Phi = \left(k \sum_{j\ne i} \frac{P^{(ij)}}{x_i-x_j} +
\lambda^{(i)}
  \right) \Phi, \qquad i=1,2,\ldots,N,               \label{eq:kz}$$ where $\Phi = \Phi(x_1,x_2,\ldots,x_N)$ takes values in the tensor product $V\otimes V\otimes\cdots\otimes V = V^{\otimes N}$ of some $N$-dimensional vector space $V$, $P^{(ij)}$ is the permutation of the $i^{\rm th}$ and $j^{\rm
th}$ factors, $\lambda = \hbox{\rm diag}(\lambda_1, \ldots, \lambda_N)$ is a diagonal matrix considered as a parameter, $\lambda^{(i)}$ is the operator in $V^{\otimes N}$ acting as $\lambda$ on the $i^{\rm th}$ factor and identically on all other factors.

Matsuo [@mat] and Cherednik [@che] indeed showed that if one considers solutions of (\[eq:kz\]) that can be written as $$\Phi = \sum_{\sigma\in S_N} {\Phi_{\sigma}}e_{\sigma}, \qquad e_{\sigma}
    = e_{\sigma(1)} \otimes e_{\sigma(2)} \otimes \cdots \otimes
    e_{\sigma(N)}, \label{eq:Phi}$$ where $S_N$ is the symmetric group, and $e_k$ denotes a column vector with entry 1 in row $k$ and zeroes everywhere else, then the symmetric and antisymmetric combinations $$\varphi^{[N]} = \sum_{\sigma \in S_N} \Phi_{\sigma}, \qquad \varphi^{[1^N]} =
  \sum_{\sigma \in S_N} (-1)^{\sigma} \Phi_{\sigma},   \label{eq:sa}$$ of the components $\Phi_{\sigma}$ of function (\[eq:Phi\]), where $(-1)^{\sigma}$ denotes the parity of permutation $\sigma$, are eigenfunctions with eigenvalue $- \sum_i \lambda_i^2$ of the Calogero Hamiltonian, $$\left(- \Delta + V_C^{(k)} + \sum_i \lambda_i^2\right) \varphi^{[N]} =
  \left(- \Delta + V_C^{(k+1)} + \sum_i \lambda_i^2\right) \varphi^{[1^N]} = 0,
  \label{eq:hcal}$$ corresponding to parameters $k$ and $k+1$, respectively.

In the present communication, we report on some new results [@cq] extending those of Matsuo and Cherednik to some CSM systems with exchange terms, thereby providing new links between the latter and some generalized KZ equations. Such equations are reviewed in Sec. 2, and used in Sec. 3 to construct wave functions for CSM systems with exchange terms. Section 4 contains the conclusion.

Generalized Knizhnik-Zamolodchikov equations
============================================

Let us generalize system (\[eq:kz\]) into $$\partial_i\Phi = \left(\sum_{j\ne i} \left(f_{ij}(x_i-x_j) P^{(ij)} +
    c\, T^{(ij)}\right) + \lambda^{(i)}\right) \Phi, \qquad i=1,2,\ldots,N,
    \label{eq:genkz}$$ where all symbols keep the same meaning as in (\[eq:kz\]), but $k
(x_i-x_j)^{-1}$ is replaced by a yet undetermined function $f_{ij}(x_i-x_j)$, and there is an additional term proportional to an operator $T^{(ij)}$ acting only on the $i^{\rm th}$ and $j^{\rm th}$ factors, and such that $T$ is the following operator on $V\otimes
V$: $$T = \sum_{k>l} \left(E_{kl}\otimes E_{lk} - E_{lk}\otimes E_{kl}\right).
    \label{eq:T}$$ Here $E_{kl}$ denotes the $N\times N$ matrix with entry 1 in row $k$ and column $l$ and zeroes everywhere else (for a previous use of the operator $T^{(ij)}$, see e.g. [@fel] and references therein).

Let us again restrict ourselves to solutions of (\[eq:genkz\]) that can be written in the form given by (\[eq:Phi\]). For such functions, Eq. (\[eq:genkz\]) is equivalent to the set of equations $$\partial_i{\Phi_{\sigma}}= \sum_{j\ne i} \left(f_{ij}(x_i-x_j) + c\, {\tau^{(ij)}_{\sigma}}\right)
  \Phi_{\sigma \circ p_{ij}} + {\lambda_{\sigma(i)}} {\Phi_{\sigma}}, \qquad i=1,2,\ldots,N,
\label{eq:genkzcom}$$ where $\sigma$ is an arbitrary permutation of $S_N$, $p_{ij}$ denotes the transposition of $i$ and $j$, and ${\tau^{(ij)}_{\sigma}} \equiv \hbox{\rm sgn}\bigl(\sigma(i)
-
\sigma(j)\bigr)$ satisfies the relations $$\begin{aligned}
  & & {\tau^{(ij)}_{\sigma}} = - \tau^{(ij)}_{\sigma \circ p_{ij}} = - {\tau^{(ji)}_{\sigma}}, \qquad
      \tau^{(ik)}_{\sigma \circ p_{ij}} = {\tau^{(jk)}_{\sigma}}, \qquad
      \tau^{(kl)}_{\sigma \circ p_{ij}} = {\tau^{(kl)}_{\sigma}}, \nonumber \\
  & & {\tau^{(ij)}_{\sigma}} {\tau^{(ik)}_{\sigma}} + {\tau^{(jk)}_{\sigma}} {\tau^{(ji)}_{\sigma}} + {\tau^{(ki)}_{\sigma}} {\tau^{(kj)}_{\sigma}} = 1,
      \label{eq:tau}\end{aligned}$$ for any $i\ne j \ne k \ne l$. In deriving Eq. (\[eq:genkzcom\]), use has been made of the properties that the operators $P^{(ij)}$, $T^{(ij)}$, and $\lambda^{(i)}$ transform the components ${\Phi_{\sigma}}$ into $\Phi_{\sigma \circ p_{ij}}$, ${\tau^{(ij)}_{\sigma}} \Phi_{\sigma
\circ
p_{ij}}$, and ${\lambda_{\sigma(i)}} {\Phi_{\sigma}}$, respectively.

It can be easily shown [@cq] that the integrability conditions of (\[eq:genkzcom\]), i.e., $\partial_j \partial_i \Phi_{\sigma} = \partial_i
\partial_j \Phi_{\sigma}$ for any $i$, $j=1$, 2, $\ldots$, $N$, and any $\sigma
\in
S_N$, are satisfied if and only if $$f_{ij}(u) = - f_{ji}(-u), \label{eq:ic1}$$ for any $i$, $j$, such that $i<j$, and $$f_{ij}(u) f_{jk}(v) - f_{ik}(u+v) \left[f_{ij}(u) + f_{jk}(v)\right] =
  - c^2, \label{eq:ic2}$$ for any $i$, $j$, $k$, such that $1\le i<j<k\le N$.

Equation (\[eq:ic2\]) looks like a functional equation first considered by Sutherland [@suth], and solved by Calogero [@cal3] through a small-$x$ expansion. A similar procedure can be used here to derive all the solutions of (\[eq:ic2\]) that are odd and meromorphic in a neighbourhood of the origin [@cq]. Among the latter, one finds $$f_{ij}(u) = f_{ji}(u) = F(u), \qquad 1\le i<j\le N, \label{eq:mero}$$ where $F(u)$ denotes the function $$F(u) = \cases{
      k \omega \coth \omega u & if $c^2 = k^2\omega^2 >0$, \cr
      \noalign{\smallskip}
      k/u                     & if $c^2 = 0$, \cr
      \noalign{\smallskip}
      k \omega \cot \omega u  & if $c^2 = -k^2\omega^2 <0$, \cr}
  \label{eq:F}$$ and $N$ may take any value such that $N\ge3$. For $c^2=0$, function (\[eq:mero\]) is the unique solution of Eq. (\[eq:ic2\]) that is odd and meromorphic in a neighbourhood of the origin, while for $c^2\ne0$, there is another solution, for which not all $f_{ij}(u)$’s are equal, and which will not play any role in the next Section.

Eq. (\[eq:ic2\]) also has some singular solutions, such as $$f_{ij}(u) = f_{ji}(u) = c\, \mbox{sgn}(u) = c\, [\theta(u) - \theta(-u)],
  \qquad 1\le i<j\le N, \label{eq:sing}$$ where $\theta(u)$ denotes the Heaviside function.

Generalization of the result of Matsuo and Cherednik
====================================================

From a set of $N!$ functions ${\Phi_{\sigma}}(x_1,\ldots,x_N)$, $\sigma \in
S_N$, satisfying Eq. (\[eq:genkzcom\]), one can construct in general $N!$ functions ${\varphi^{[f]}_{rs}}(x_1,\ldots,x_N)$, defined by $${\varphi^{[f]}_{rs}} = \sum_{\sigma\in S_N} {V^{[f]}_{rs}(\sigma)} {\Phi_{\sigma}}, \label{eq:mixed}$$ where $[f] \equiv \left[f_1 f_2 \ldots f_N\right]$ runs over all $N$-box Young diagrams, $r$ and $s$ label the standard tableaux associated with $[f]$, arranged in lexicographical order, and ${V^{[f]}_{rs}(\sigma)}$ denotes Young’s orthogonal matrix representation of $S_N$ [@ruth]. For $[f] = [N]$ or $\bigl[1^N\bigr]$, since $V^{[N]}(\sigma) = 1$ and $V^{[1^N]}(\sigma) =
(-1)^{\sigma}$, functions (\[eq:mixed\]) reduce to those considered by Matsuo and Cherednik and given in (\[eq:sa\]).

By using the previous equations, as well as some elementary results about transpositions, $$p_{ik} \circ p_{ij} = p_{ij} \circ p_{jk} = p_{jk} \circ p_{ik},
  \label{eq:trans}$$ and matrix representations, $${V^{[f]}_{rs}(\sigma \circ \sigma')} = \sum_t {V^{[f]}_{rt}(\sigma)} {V^{[f]}_{ts}(\sigma')},
  \qquad {V^{[f]}_{rs}(1)} = \delta_{r,s}, \label{eq:matrep}$$ it is straightforward to show [@cq] that the functions ${\varphi^{[f]}_{rs}}$ satisfy the system of equations $$\begin{aligned}
  \partial_i {\varphi^{[f]}_{rs}} & = & \sum_{j\ne i} f_{ij} \sum_t {\varphi^{[f]}_{rt}} {V^{[f]}_{ts}(p_{ij})}
     - c \sum_{j\ne i} \sum_t \biggl(\sum_{\sigma} {\tau^{(ij)}_{\sigma}} {V^{[f]}_{rt}(\sigma)} {\Phi_{\sigma}}\biggr) {V^{[f]}_{ts}(p_{ij})} \nonumber \\
  & & \mbox{} + \sum_{\sigma} {\lambda_{\sigma(i)}} {V^{[f]}_{rs}(\sigma)} {\Phi_{\sigma}}, \qquad
     i=1,2,\ldots,N, \label{eq:first}\end{aligned}$$ and that their Laplacian is given by $$\Delta {\varphi^{[f]}_{rs}} = \left({\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\left(f_{ij}^2(x_i-x_j) + \left(\partial_i
  f_{ij}(x_i-x_j)\right) K_{ij} - c^2\right) + \sum_i \lambda_i^2\right)
  {\varphi^{[f]}_{rs}}. \label{eq:laplacian}$$ In (\[eq:laplacian\]), $K_{ij} = K_{ji}$, $1\le i<j\le N$, are some operators, whose action on ${\varphi^{[f]}_{rs}}$ is defined by $$K_{ij} {\varphi^{[f]}_{rs}} = \sum_t {\varphi^{[f]}_{rt}} {V^{[f]}_{ts}(p_{ij})}. \label{eq:K}$$

In the special cases where $[f] = [N]$ or $\bigl[1^N\bigr]$, the operators $K_{ij}$ behave as $I$ or $-I$, respectively. Hence, for $f_{ij}$ given by (\[eq:mero\]) and (\[eq:F\]), where $c^2=0$, Eq. (\[eq:laplacian\]) reduces to Eq. (\[eq:hcal\]), i.e., Matsuo and Cherednik’s result. Let us emphasize that Eq. (\[eq:laplacian\]) is also valid in the mixed-symmetry cases, and for any function ${\varphi^{[f]}_{rs}}$ constructed from any solution of (\[eq:genkzcom\]) via transformation (\[eq:mixed\]).

As for $[f] \ne [N]$, $\bigl[1^N\bigr]$, the operators $K_{ij}$ have a rather complicated effect on the functions ${\varphi^{[f]}_{rs}}$, it is convenient to restrict the latter so that for any $i<j$, $K_{ij}$ may be interpreted as a permutation operator acting on the variables $x_i$ and $x_j$, and leaving the remaining variables $x_k$ unchanged, $$K_{ij} x_j = x_i K_{ij}, \qquad K_{ij} x_k = x_k K_{ij} \qquad k\ne i,j.
  \label{eq:xpermute}$$ From (\[eq:K\]), it results that the conditions to be fulfilled by ${\varphi^{[f]}_{rs}}$ are $$\begin{aligned}
  \lefteqn{{\varphi^{[f]}_{rs}}(x_1,\ldots,x_j,\ldots,x_i,\ldots,x_N)} \nonumber \\
  & = &\sum_t {\varphi^{[f]}_{rt}}(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_N) {V^{[f]}_{ts}(p_{ij})},
      \qquad 1\le i<j\le N. \label{eq:condphi}\end{aligned}$$ In terms of the components ${\Phi_{\sigma}}$ of (\[eq:Phi\]), such conditions amount to $$\begin{aligned}
  \lefteqn{{\Phi_{\sigma}}(x_1,\ldots,x_j,\ldots,x_i,\ldots,x_N)} \nonumber \\
  & = & \Phi_{\sigma\circ p_{ij}}(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_N),
      \qquad 1\le i<j\le N, \label{eq:condPhi}\end{aligned}$$ for any $\sigma\in S_N$. By differentiating both sides of (\[eq:condPhi\]) with respect to $x_k$ and using (\[eq:genkzcom\]) to calculate the derivatives, one finds that Eqs. (\[eq:genkzcom\]) and (\[eq:condPhi\]) are compatible if and only if all functions $f_{ij}(u)$, $i\ne j$, coincide, hence in cases such as (\[eq:mero\]) and (\[eq:sing\]).

For instance, when $f_{ij}$ is given by (\[eq:mero\]) and (\[eq:F\]), where $c^2>0$ (hyperbolic case), Eq. (\[eq:laplacian\]) becomes $$\left(- \Delta + \omega^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\left({\mathop{\rm csch}\nolimits}\omega(x_i-x_j)\right)^2
  k(k-K_{ij}) + \sum_i \lambda_i^2\right) {\varphi^{[f]}_{rs}} = 0, \label{eq:suthex}$$ with $K_{ij}$ defined by (\[eq:xpermute\]). Such an equation (as well as similar results for the remaining cases) shows that from any solution of type (\[eq:Phi\]), (\[eq:condPhi\]) of the KZ equations (\[eq:genkz\]), with $f_{ij}$ given in (\[eq:mero\]), one can obtain eigenfunctions ${\varphi^{[f]}_{rs}}$ of the CSM Hamiltonians [@cal1]–[@mos] with exchange terms [@ber], which are characterized by any given permutational symmetry $[f]$ under particle coordinate exchange. To obtain wave functions describing an $N$-boson (resp.$N$-fermion) system, it only remains to combine ${\varphi^{[f]}_{rs}}$ with a spin function transforming under the same (resp. conjugate) irreducible representation $[f]$ (resp. $[\tilde f]$) under exchange of the spin variables. A similar result is valid for the Hamiltonian with delta-function interactions [@lieb], corresponding to the functions $f_{ij}$ given in (\[eq:sing\]).

Conclusion
==========

By extending the Matsuo and Cherednik results to some integrable models with exchange terms and wave functions of any permutational symmetry, we showed that there exists a strong interplay between such models and (generalized) KZ equations. As already noted by Brink and Vasiliev [@brink2], this also hints at some deep relationship between the latter and Dunkl operators.

Whether some results similar to those presented here also hold true for elliptic CSM models, and for integrable models related to root systems of algebras different from ${\cal A}_{N-1}$ [@pere], remains an interesting open question.

[99]{}

Calogero F.: J. Math. Phys. [*10*]{} (1969) 2191, 2197; ibid [*12*]{} (1971) 419.

Calogero F., Ragnisco O., and Marchioro C.: Lett. Nuovo Cimento [*13*]{} (1975) 383.\
Calogero F.: Lett. Nuovo Cimento [*13*]{} (1975) 411.

Calogero F.: Lett. Nuovo Cimento [*13*]{} (1975) 507.

Sutherland B.: Phys. Rev. A [*4*]{} (1971) 2019; ibid [*5*]{} (1972) 1372; Phys. Rev. Lett. [*34*]{} (1975) 1083.

Moser J.: Adv. Math. [*16*]{} (1975) 1.

Olshanetsky M. A. and Perelomov A. M.: Phys. Rep. [*94*]{} (1983) 313.

Polychronakos A. P.: Phys. Rev. Lett. [*69*]{} (1992) 703.

Brink L., Hansson T. H., and Vasiliev M. A.: Phys. Lett. B [*286*]{} (1992) 109.

Dunkl C. F.: Trans. Am. Math. Soc. [*311*]{} (1989) 167.

Bernard D., Gaudin M., Haldane F. D. M., and Pasquier V.: J. Phys. A [*26*]{} (1993) 5219.

Knizhnik V. G. and Zamolodchikov A. B.: Nucl. Phys. B [*247*]{} (1984) 83.

Matsuo A.: Invent. Math. [*110*]{} (1992) 95.

Cherednik I. V.: Integration of Quantum Many-Body Problems by Affine KZ Equations. Preprint Kyoto RIMS (1991).

Quesne C.: J. Phys. A [*28*]{} (1995) in press.

Felder G. and Veselov A. P.: Commun. Math. Phys. [*160*]{} (1994) 259.

Rutherford D. E.: Substitutional Analysis. Edinburgh UP, Edinburgh, 1948.

Lieb E. H. and Liniger W.: Phys. Rev. [*130*]{} (1963) 1605.\
Yang C. N.: Phys. Rev. Lett. [*19*]{} (1967) 1312; Phys. Rev. [*168*]{} (1968) 1920.

Brink L. and Vasiliev M. A.: Mod. Phys. Lett. A [*8*]{} (1993) 3585.

[^1]: Presented at the 4th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 1995