\documentclass[ALCO, Unicode]{cedram} \usepackage[matrix,arrow,tips,curve]{xy} \usepackage{wrapfig} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{tabu} \usepackage{comment} \usepackage{epstopdf} \usepackage{mathtools} \usepackage{subcaption} \usepackage{multicol} \usepackage{verbatim} \usepackage{eurosym} \usepackage{apptools} \usepackage{tabularx} \usepackage{enumitem} \usepackage{bbm} \newcommand{\eps}{\varepsilon} \newcommand{\A}{\mathbf A} \newcommand{\B}{\mathbf B} \newcommand{\V}{\mathbf V} \newcommand{\W}{\mathbf W} \newcommand{\w}{\mathbf w} \newcommand{\x}{\mathbf x} \newcommand{\y}{\mathbf y} \newcommand{\ba}{\boldsymbol\alpha} \newcommand{\bb}{\boldsymbol\beta} \newcommand{\be}{\boldsymbol \eta} \newcommand{\bz}{\boldsymbol \zeta} \newcommand{\bg}{\boldsymbol \gamma} \newcommand{\s}{\mathfrak s} \newcommand{\p}{\mathfrak p} \newcommand{\q}{\mathfrak q} \newcommand{\R}{\mathcal R} \newcommand{\J}{\mathbf J} \newcommand{\ii}{\mathbf i} \newcommand{\dd}{\mathrm d} \newcommand{\E}{\mathbb E} \newcommand{\T}{\mathsf T} \newcommand{\D}{\mathcal D} \newcommand{\PP}{\mathcal P} \newcommand{\la}{\longrightarrow} \title{Eigenvalues of Heckman--Polychronakos operators} \author{\firstname{Charles} \lastname{Dunkl}} \address[Charles Dunkl]{University of Virginia\\ Department of Mathematics\\ Charlottesville, VA 22904 (USA) } \email{cfd5z@virginia.edu} \author{\firstname{Vadim} \lastname{Gorin}} \address[Vadim Gorin]{University of California at Berkeley\\ Departments of Statistics and Mathematics\\ Berkeley, CA 94720 (USA) } \email{vadicgor@gmail.com} \keywords{Jack polynomial, operators on polynomials, characters of representations, symmetric group} \subjclass{05E05, 05E10, 20C30, 15A18, 81Q05} \begin{document} \begin{abstract} Heckman--Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal{D}_i$ as $\mathcal{P}_m= \sum_{i=1}^N (x_i \mathcal{D}_i)^m$. They have been known since 1990s in connection with trigonometric Calogero--Moser--Sutherland Hamiltonian and Jack symmetric polynomials. We explicitly compute the eigenvalues of these operators for symmetric and skew-symmetric eigenfunctions, as well as partial sums of eigenvalues for general polynomial eigenfunctions. \end{abstract} \maketitle \theoremstyle{plain} \newtheorem{examples}[cdrthm]{Examples} \section{Introduction} Our paper is motivated by two directions. On one hand, the theory of \emph{quantum integrable systems}, or more generally \emph{spectral theory}, seeks to construct families of commuting operators and to understand the structure of their eigenvalues and eigenfunctions. On the other hand, \emph{integrable probability} often exploits such operators to extract information about stochastic systems and their asymptotic behavior. We investigate a family of differential-difference operators acting on polynomials in $N$ variables $x_1,\dots,x_N$, known as the \emph{Heckman--Polychronakos operators}, which were first introduced in~\cite{heckman1991elementary,polychronakos1992exchange} in connection with the study of the (trigonometric) Calogero--Moser--Sutherland Hamiltonian. The key properties of interest for these operators were that they commute with each other and with the Calogero--Moser--Sutherland Hamiltonian, and that Jack symmetric polynomials (whose extensive theory was developed just a couple of years before that, see~\cite{stanley1989some}, \cite[Chapter VI]{macdonald1998symmetric}) are their eigenfunctions. Shortly after their introduction, two important developments occurred. First,~\cite{cherednik1991unification,cherednik1994integration} introduced an alternative family of \emph{Cherednik operators} satisfying the same three properties. Second, the theory was lifted to the level of Macdonald polynomials, of which Jack polynomials are a degeneration (cf.\ \cite[Chapter 6]{macdonald1998symmetric}). In the following years Cherednik operators and their Macdonald generalizations played central roles in many developments, while the Heckman--Polychronakos operators received significantly less attention (cf.~\cite{chalykh2024dunkl} for a recent historical overview). As a result, even the most basic properties of the latter operators, such as explicit formulas for their eigenvalues, remain poorly understood. The goal of this paper is to help fill this gap in the literature and to reinitiate the study of the Heckman--Polychronakos operators. Some relationships between Cherednik and Heckman--Polychronakos operators are discussed in \cite[Section 5]{sergeev2015jack}. Both families of operators can be constructed via Newton power sums of certain ``basic'' operators. However, these basic operators exhibit different properties in the two theories: in Cherednik theory, the basic operators commute with each other, but their definition is not compatible with the natural action of the symmetric group $S_N$ on $(x_1, \dots, x_N)$; in Heckman--Polychronakos theory, the basic operators are defined in an $S_N$-equivariant way, but at the cost of losing commutativity.\footnote{We emphasize that although the basic operators $x_i\D_i$ do not commute, the Heckman--Polychronakos operators themselves do commute; see Definition~\ref{Definition_HP_operator}, Lemma~\ref{Lemma_commute}, and~\eqref{eq_commutator}.} The two parallel choices of basic operators in Macdonald $(q,t)$ theory are further discussed in~\cite{nazarov2019cherednik}, especially in Section~2.4. In the semiclassical limit,\footnote{This limit is related to the degeneration of a Lie group into a Lie algebra and to the elementary transition $\lim\limits_{\eps \to 0}(1+\eps x)^{\eps^{-1}y} = \exp(xy)$.} Jack polynomials degenerate into multivariate Bessel functions (see, e.g., \cite[Section 4]{okounkov1997shifted}), and the operators we study degenerate in parallel to objects related to the rational (rather than trigonometric) Calogero--Moser--Sutherland Hamiltonian. Interestingly, in this limit the Cherednik and Heckman--Polychronakos operators merge into a single family: their basic operators converge to the Dunkl operators of~\cite{dunkl1989differential}, which both commute and are $S_N$-equivariant. Switching to the integrable probability side, recent works~\cite{benaych2022matrix,xu2023rectangular,GXZ2024,KX2024} have made intensive use of power sums of Dunkl operators in the study of $\beta$-ensembles in random matrix theory. These studies address problems such as the addition of matrices, corner cutting, and the analysis of classical random matrix ensembles. Typical questions in this area concern the asymptotic behavior of individual eigenvalues, as well as more intricate functions of eigenvalues that capture their joint distribution as the matrix size tends to infinity. For the operators to be useful in such asymptotic analyses, two features are particularly important. First, the operators should have reasonably simple combinatorial expressions, allowing their action on explicit functions to be tractable in asymptotic regimes. Second, the eigenvalues of the operators correspond to observables of the stochastic system of interest, and thus should also be reasonably explicit. The power sums of Dunkl operators satisfy both of these criteria. The random matrix distributions discussed above have natural discrete analogues, extensively studied in~\cite{kerov2000anisotropic, borodin2005z, gorin2015multilevel,dolega2016gaussian,borodin2017gaussian, dolega2019gaussian, guionnet2019rigidity,huang2021law,dimitrov2022asymptotics,cuenca2023universality,moll2023gaussian, dimitrov2024global,gorin2024dynamical}, and closely connected to Jack polynomials. Many questions posed for matrix $\beta$-ensembles have discrete counterparts, and similar techniques may be applied to analyze them. This, however, requires lifting the power sums of Dunkl operators to the discrete setting --- leading naturally to either Cherednik or Heckman--Polychronakos operators. It is not yet clear which of the two classes is better suited for asymptotic analysis, and ideally, one would like to explore both. The desire to study asymptotic properties of these stochastic systems was the original motivation behind our investigation of Heckman--Polychronakos operators. However, their theory turned out to be far richer than expected, revealing many additional structural features and open questions that remain to be explored. The rest of the paper is organized as follows. In Section~\ref{Section_HP_operators} we formally define the Heckman--Polychronakos operators. In Section~\ref{Section_sym_eigenfunction} we present a formula for their eigenvalues on symmetric eigenfunctions. In Section~\ref{Section_classification} we propose a classification of general polynomial eigenfunctions according to the degrees in their leading monomials and the type of representation of the symmetric group $S_N$ to which these monomials belong. In Section~\ref{Section_sums} we relate partial sums of eigenvalues to values of the characters of the symmetric group and explain how this can be used for explicit computations of some of the eigenvalues. In Section~\ref{Section_N3} we compute a three variable example highlighting that in some cases the eigenvalues are much more complicated. Finally, Appendix~\ref{Section_appendix} is an appendix, where we recall various useful properties of the Dunkl operators. \section{Heckman--Polychronakos operators} \label{Section_HP_operators} Throughout the paper we fix $N=1,2,\dots$ and use the ring $\mathbb C_N=\mathbb C[x_1,x_2\dots,x_N]$ of polynomials in $N$ variables. We let $\theta$ be an auxiliary parameter which will either play the role of a formal variable or a positive number throughout the paper. \begin{defi} \label{Definition_Dunkl} For $1\le i\le N$ the \emph{Dunkl operator} $\D_i$ is a linear operator in $\mathbb C_N$ acting by \begin{equation} \D_i= \frac{\partial}{\partial x_i}+\theta \sum_{j\ne i} \frac{1-(i,j)}{x_i-x_j}, \end{equation} where $(i,j)$ stands for the operator permuting the variables $x_i$ and $x_j$. \end{defi} \begin{defi} \label{Definition_HP_operator} For $m\ge 1$ the \emph{Heckman--Polychronakos operator} $\PP_m$ is a linear operator in $\mathbb C_N$ acting by \begin{equation} \PP_m= \sum_{i=1}^N (x_i \D_i)^m, \end{equation} where $x_i$ stands for the operator of multiplication by $x_i$. \end{defi} The operators $\PP_m$ were first introduced and discussed in~\cite{heckman1991elementary,polychronakos1992exchange}. The interest in them is based on three properties, whose proofs we recall in the Appendix: \begin{itemize} \item $\PP_m$ commute with each other as operators in $\mathbb C_N$: $\PP_m \PP_k=\PP_k\PP_m$. \item When restricted onto the subring $\Lambda_N\subset \mathbb C_N$ of symmetric polynomials, the operator $\PP_2$ is the (conjugated) Calogero--Moser--Sutherland Hamiltonian. \item Symmetric Jack polynomials $J_\lambda(x_1,\dots,x_N;\, \theta)$, $\lambda_1\ge\lambda_2\ge \dots\ge \lambda_N\ge 0$, are eigenfunctions of all $\PP_m$; see~\cite{stanley1989some}, \cite[Chapter VI]{macdonald1998symmetric} for general introduction to Jack polynomials and note that it uses the parameter $\alpha=1/\theta$. \end{itemize} In contrast to another famous family of operators satisfying these three properties --- power sums of the (Jack versions of) Cherednik operators~\cite{cherednik1991unification,cherednik1994integration} --- the individual terms $(x_i \D_i)^m$ \emph{do not} commute.\footnote{For a fruitful study of commutators of $\D_i$ and $x_j$ operators see~\cite{feigin2015dunkl}.} On the other hand, the advantage of Heckman--Polychronakos operators is that their definition is simpler than the Cherednik ones and they are equivariant with respect to the symmetric group action. The aim of this paper is to study the spectrum of the operators $\PP_m$: we would like to understand the eigenvalues corresponding to Jack polynomials $J_\lambda$, as well as those corresponding to other \emph{non-symmetric} eigenfunctions. The $m=1$ operator is simpler than others: \begin{multline*} \sum_{i=1}^N x_i \D_i = \sum_{i=1}^N x_i \frac{\partial}{\partial x_i}+\theta \sum_{1\le i\gamma_i} (i,j) x^{\gamma}, \qquad 1\le i \le N. \end{equation} \begin{lemma} \label{Lemma_triangularity} For any partition $\lambda$ and any degree sequence $\gamma$ with $\gamma^+=\lambda$, we have {\small $$ \PP_m[x^{\gamma}]=(T_1^m+T_2^m+\dots+T_N^m)[x^{\gamma}]+(\text{linear combination of monomials } x^{\tilde \gamma}\text{ with }\tilde \gamma^+\prec\lambda). $$} \end{lemma} \begin{proof} Note that \begin{multline} x_i \frac{1-(i,j)}{x_i-x_j} \bigl[x_1^{\gamma_1},\dots, x_N^{\gamma_N}\bigr]= \prod_{a\ne i,j} x_a^{\gamma_a} \cdot x_i \frac{x_i^{\gamma_i} x_j^{\gamma_j}-x_i^{\gamma_j} x_j^{\gamma_i}}{x_i-x_j}\\ = \prod_{a\ne i,j} x_a^{\gamma_a}\cdot \begin{cases} x_i^{\gamma_i} x_j^{\gamma_j}+ x_i^{\gamma_i-1} x_j^{\gamma_j+1}+\dots+ x_i^{\gamma_j+1} x_j^{\gamma_i-1},& \gamma_j<\gamma_i,\\ -x_i^{\gamma_j} x_j^{\gamma_i}- x_i^{\gamma_j-1} x_j^{\gamma_i+1}-\dots- x_i^{\gamma_i+1} x_j^{\gamma_j-1},& \gamma_j>\gamma_i,\\ 0,& \gamma_j=\gamma_i. \end{cases} \end{multline} Hence, $$ x_i \D_i [x^{\gamma}]= T_i x^{\gamma}+ (\text{linear combination of monomials } x^{\tilde \gamma}\text{ with }\tilde \gamma^+\prec\gamma^+). $$ Iterating the last identity for $\PP_m=\sum_{i=1}^N (x_i \D_i)^m$, we arrive at the statement of the lemma. \end{proof} \begin{lemma} \label{Lemma_self_adjoint} Consider a scalar product on $\mathbb C_N$, where $\overline{g}$ stands for complex conjugation and $\theta\ge 0$: {\small \begin{equation} \label{eq_scalar_product1} \langle f,g \rangle =\int_{0}^{2\pi}\cdots\int_{0}^{2\pi} f\bigl(e^{\ii \phi_1},\dots,e^{\ii\phi_N}\bigr) \overline{g\bigl(e^{\ii \phi_1},\dots,e^{\ii\phi_N}\bigr)} \prod_{1\le i\gamma_{j_{k-2}}>\dots>\gamma_{j_1}>\gamma_1$. The transpositions $(1,j_a)$ correspond to the second term in~\eqref{eq_x4} and the ordering appears because the operator $T_1$ can only increase the degree of $x_1$, but never decrease. On the other hand, the prefactor $p_{k;q}$ collects all the factors coming from the first term; it is a polynomial in various $\theta$-shifted $\gamma_i$ of degree $(m+1-k)$, which depends on the collection $\gamma_1, \gamma_{j_1},\dots,\gamma_{j_{k-1}}$ and on $\theta$. Note also that the product of $k-1$ transpositions becomes a $k$-cycle $$ (1,j_{k-1})(1, j_{k-2})\cdots (1,j_{1})=(1, j_{1}, j_{2},\dots, j_{k-1}). $$ We further represent $g$ in~\eqref{eq_projector} as $$ g=\tilde g\cdot (1,j_1) \cdots (1,j_{k-2})(1, j_{k-1}), $$ and note that when we plug into $\mathrm{Trace}_{V_\lambda} \left[\pi_\tau T_1^m\right]$, and multiply the sums, only the terms with $\tilde g=\mathrm{Id}$ give non-zero contributions. We also note that the trace is a central function and its values on all $k$-cycles are the same. Therefore, \begin{multline} \label{eq_x6} N \mathrm{Trace}_{V_\lambda} \left[\pi_\tau T_1^m\right]\\ = \frac{\dim \tau}{(N-1)!} \sum_{k=1}^{m+1} (-\theta)^{k-1} \chi^\tau\bigl( (1,2,\dots,k) \bigr) \sum_{\gamma} \sum_{j_1,\dots,j_{k-1}} p_{k;m}[\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}]. \end{multline} It remains to compute the last double sum and we claim that \begin{multline} \label{eq_x11} \frac{1}{(N-1)!}\sum_{\gamma} \sum_{j_1,\dots,j_{k-1}} p_{k;m}[\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}]\\= \text{Coefficient of }(-\theta)^{k-1}\text{ in: } \begin{pmatrix}1&1&\dots&1\end{pmatrix} \cdot \begin{pmatrix} \ell_1& -\theta &\dots & && -\theta \\ 0 & \ell_2& -\theta &&& -\theta\\ 0 & 0 & \ell_3 & -\theta && \vdots\\ \vdots &\vdots &\vdots&&& -\theta\\ 0 &0 &&\dots &0 &\ell_N \end{pmatrix}^m \cdot \begin{pmatrix} 1\\ 1 \\ \vdots \\ 1 \end{pmatrix} \end{multline} Indeed, there are $N!$ terms in the sum over $\gamma$, however, only the rank of $\gamma_1$ among all coordinates of $\gamma$ matters for the computation of the following $ \sum_{j_1,\dots,j_{k-1}}$. Hence, $\frac{1}{(N-1)!}\sum_{\gamma}$ can be interpreted as a sum with $N$ terms corresponding to $N$ possible ranks of $\gamma_1$. Next, we notice that the first term in~\eqref{eq_x4} is always equal to $\ell_a$ and therefore matches the diagonal element of the matrix in~\eqref{eq_x11}. Similarly, the $-\theta(i,j)$ terms in~\eqref{eq_x4} match the off-diagonal $(-\theta)$ elements in the matrix in~\eqref{eq_x11}. Hence, we see that the combinatorics of rising the matrix to the $m$-th power in~\eqref{eq_x11} and raising $T_i$ to the $m$-th power is exactly the same. Hence, the identity~\eqref{eq_x11}. It remains to identify the right-hand side of~\eqref{eq_x11} with $h^{(k)}_{m+1-k}\bigl(\ell_1,\dots,\ell_N\bigr)$, which is immediate from the definition~\eqref{eq_x3}: the $k$-element subset $\{i_1,\dots,i_k\}$ encodes those diagonal elements of the matrix in~\eqref{eq_x11}, which enter into the computation of the $m$-th power. Finally, the summation $\sum_{k=1}^{m+1}$ in~\eqref{eq_x6} can be restricted to $\sum_{k=1}^{\min(m+1,N)}$ in~\eqref{eq_distinct_trace}, because for $k>N$ finding $k$ distinct coordinates $\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}$ becomes impossible and the corresponding sum in~\eqref{eq_x6} is empty. \end{proof} \subsection{General degrees} We now proceed to the most general result, in which parts of $\lambda$ are allowed to have multiplicities. Given an integer $1\le p\le N$, a multiplicity composition (i.e.\ a sequence of $p$ positive integers) $\mathbf n=(n_1, n_2, \dots, n_p>0)$ with $\sum_{i=1}^p n_i=N$, and a degree sequence $\mathbf d=(d_1> d_2> \dots> d_p\ge 0)$, we introduce a partition \begin{equation} \label{eq_general_lambda} \lambda= \underbrace{d_1,\dots, d_{1}}_{n_1}, \underbrace{d_2,\dots, d_{2}}_{n_2}, \dots, \underbrace{d_p,\dots, d_{p}}_{n_p}. \end{equation} For any $k$-element subset $\mathcal A\subset \{1,\dots,p\}$ of the form $\{a_1\gamma_{j_{k-2}}>\dots>\gamma_{j_{1}}>\gamma_1$. We again represent $g$ as $$ g=\tilde g\cdot (1,j_{1}) \cdots (1,j_{k-2})(1, j_{k-1}), $$ and arrive at a generalization of~\eqref{eq_x6}: \begin{multline} \label{eq_x16} \sum_{i=1}^{\dim V_{\lambda,\tau}} \mathrm{eig}_m(\lambda,\tau,i) =N \mathrm{Trace}_{V_{\lambda}} \Bigl( \pi_\tau (T_1)^m\Bigr)= \frac{\dim \tau}{(N-1)!} \sum_{k=1}^{m+1} (-\theta)^{k-1} \\ \times \sum_{\gamma} \sum_{j_1,\dots,j_{k-1}} \sum_{\tilde g\in S_N\mid \tilde g\gamma=\gamma} \chi^\tau\bigl(\tilde g \cdot (1,j_{1}) \cdots (1,j_{k-2})(1, j_{k-1}) \bigr) p_{k;m}[\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}] \end{multline} What remains is to carefully match the last formula with the claimed~\eqref{eq_general_trace}. First, the summation index $k$ runs from $1$ to $m+1$ in~\eqref{eq_x16}, however, for $k>p$, it is impossible to choose $k$ distinct $\gamma_{j_{k-1}}>\gamma_{j_{k-2}}>\dots>\gamma_{j_{1}}>\gamma_1$ and the sum is empty; hence, we can restrict the upper bound to be $\min(k+1,p)$. Next, the summation over all $\gamma$ involves $$ \frac{N!}{(n_1)! (n_2)!\cdots (n_p)!} $$ terms. Absorbing the factorials from the denominator into the character summation (they are present in~\eqref{eq_family_character}) and cancelling $N!$ with $(N-1)!$ in denominator in~\eqref{eq_x16}, we are left with a sum involving a single fixed $\gamma$ and $ \pi_\tau \sum_{i=1}^N (T_i)^m$. We claim that the summation over $\tilde g$ in~\eqref{eq_x16} gives the character averages of~\eqref{eq_family_character}. Indeed, note that $$ (1,j_{1}) \cdots (1,j_{k-2})(1, j_{k-1})=(j_{k-1}, j_{k-2},\dots,j_{1},1) $$ and the corresponding degrees $\gamma_{j_{k-1}}, \gamma_{j_{k-2}},\dots,\gamma_{1}$ are arranged in the decreasing order, which correspond to increasing indices of $d_i$ --- matching the cycle $\mathfrak c$ used in~\eqref{eq_family_character}. The stabilizer of $\gamma$, i.e.\ the subgroup $\{\tilde g\in S_N\mid \tilde g\gamma=\gamma\}$ is not necessarily $S_{n_1}\times S_{n_2}\times \dots S_{n_p}$, but it is isomorphic to this group, with isomorphism obtained by renaming $\{1,2,\dots,N\}$. Since the characters are central, this leads to the same function as~\eqref{eq_family_character}. It remains to identify the sums of $p_{k;m}[\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}]$. Note that when we act with $(T_i)^m$ using its definition~\eqref{eq_x4}, the factors $\left(\gamma_i + \theta \#\{j\mid \gamma_j<\gamma_i\}\right)$ correspond to various $\tilde \ell_a$. On the other hand, each application of a term from $\theta \sum_{j\mid \gamma_j>\gamma_i} (i,j) x^{\gamma}$, leads to addition of an additional index ${j_a}$. In this index, only the value of $\gamma_{j_a}$ matters. The set $\mathcal A$ in~\eqref{eq_general_trace} is identified with $\{\gamma_1,\gamma_{j_1},\dots,\gamma_{j_{k-1}}\}$ in~\eqref{eq_x16}, leading to the combinatorial factor $\prod_{a\in \mathcal A} n_a$, counting the number of ways to choose indices $\gamma_{j_a}$ corresponding to~$\mathcal A$ (the $n_a$ corresponding to $\gamma_1$ should be also accounted, as it arises from the first application of one of the operators $T_i$ in $\sum_{i=1}^N (T_i)^m$). Once the set $\mathcal A$ is fixed, the sums of products of $\tilde \ell_a$ coming from $\left(\gamma_i + \theta \#\{j\mid \gamma_j<\gamma_i\}\right)$ in $T_i$ combine precisely into $h^{\mathcal A}_{m+1-k}$ of~\eqref{eq_general_trace}. \end{proof} \begin{lemma} \label{Lemma_symmetric_ev_match} For the trivial character, $\tau=(N)$, the formulas of Theorems~\ref{Theorem_Traces_general} and~\ref{Theorem_symmetric_ef} are the same. \end{lemma} \begin{proof} The formula~\eqref{eq_Jack_ev} can be rewritten in terms of the $m$-th power of $N\times N$ matrix as \begin{equation} \label{eq_x12} \begin{pmatrix} 1 & 1 & 1 & \ldots & 1 \end{pmatrix} \begin{pmatrix} \ell_{1} & -\theta & -\theta & \ldots & -\theta\\ 0 & \ell_{2} & -\theta & \ldots & -\theta\\ 0 & 0 & \ell_{3} & \ldots & -\theta\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \ldots & \ell_{N} \end{pmatrix} ^{m} \begin{pmatrix} 1\\ 1\\ 1\\ \vdots\\ 1 \end{pmatrix}, \end{equation} where $\ell_i=\lambda_i+\theta(N-i)$. On the other hand, the formula~\eqref{eq_general_trace} can rewritten in terms of $m$-th power of $p\times p$ matrix as \begin{equation} \label{eq_x13} \begin{pmatrix} n_{1} & n_{2} & n_{3} & \ldots & n_{p} \end{pmatrix} \begin{pmatrix} \tilde \ell_{1} & -n_{2}\theta & -n_{3}\theta & \ldots & -n_{p}\theta\\ 0 & \tilde \ell_{2} & -n_{3}\theta & \ldots & -n_{p}\theta\\ 0 & 0 & \tilde \ell_{3} & \ldots & -n_{p}\theta\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \ldots & \tilde \ell_{p} \end{pmatrix} ^{m} \begin{pmatrix} 1\\ 1\\ 1\\ \vdots\\ 1 \end{pmatrix}, \end{equation} where $\tilde \ell_i$ are a subset of $\ell_i$ given by $\tilde \ell_i= d_i + \theta (n_{i+1}+n_{i+2}+\dots+n_p)$ in the notation~\eqref{eq_general_lambda}. Our task is to prove~\eqref{eq_x12}=\eqref{eq_x13}. One can do this directly, but our approach is to proceed through generating functions. On the~\eqref{eq_x12} side, the generating function was computed in Proposition~\ref{Proposition_sym_gen_funct}. Let $\widetilde{\mathrm{eig}}_m(\lambda)$ denote the eigenvalues computed by~\eqref{eq_x13}. Repeating the argument of Proposition~\ref{Proposition_sym_gen_funct}, we have \begin{multline*} 1-\theta z \sum_{m=0}^{\infty} \widetilde{\mathrm{eig}}_m(\lambda) z^m = \sum_{\begin{smallmatrix} \mathcal A\subset \{1,\dots,p\} \end{smallmatrix}} (-\theta z)^{|\mathcal A|} \prod_{a\in \mathcal A} \frac{n_a}{1-\tilde \ell_{a}z}=\prod_{i=1}^p\left(1-\frac{n_i \theta z}{1-\tilde \ell_i z}\right)\\=\prod_{i=1}^p\frac{1-(\tilde \ell_i+n_i \theta) z}{1-\tilde \ell_i z}. \end{multline*} The last expression matches the right-hand side of~\eqref{eq_x8}, once we notice the telescoping cancellations in~\eqref{eq_x8}. \end{proof} \subsection{\texorpdfstring{$(N-1,1)$}{(N-1,1)} isotype} While the formula of Theorem~\ref{Theorem_Traces_general} is very general, it produces an answer in terms of the functions $\chi^\tau[\mathcal A; \mathbf n]$. Evaluation of these functions is a separate task. As we have already mentioned right after Theorem~\ref{Theorem_Traces_general}, for $\tau=(N)$ and $\tau=1^N$ the evaluation is particularly simple. Another case where it is fully explicit is $\tau=(N-1,1)$. \begin{theorem} \label{Theorem_character_sum_fundamental} Let $\tau=(N-1,1)$ and $\mathbf n=(n_1, n_2, \dots, n_p>0)$. Then we have \begin{equation} \label{eq_N11_formula} \chi^{\tau}[\mathcal A; \mathbf n]=p-1-\sum_{a\in \mathcal A} \frac{1}{n_a}. \end{equation} \end{theorem} \begin{proof} The irreducible representation of $S_N$ of type $\tau=(N-1,1)$ is the standard (reflection) $(N-1)$-dimensional representation in the space of vectors $(x_1,\dots,x_N)$ with $\sum_{i=1}^N x_i=0$ and action $\sigma\circ (x_1,\dots,x_N)=(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(N)})$. Evaluating the character as trace, we get an expression in terms of the number of the fixed points: $$ \chi^{\tau}(\sigma)=\#\{1\le i \le N\mid \sigma(i)=i\}-1, \qquad \tau=(N-1,1). $$ Recalling the definition~\eqref{eq_family_character}, we need to compute \begin{equation} \label{eq_x18} \sum_{\tilde g\in S_{n_1}\times S_{n_2}\times \dots \times S_{n_p}}\,\, \chi^\tau(\tilde g \cdot \mathfrak c) =\sum_{\tilde g} \sum_i \mathbf 1_{ \tilde g \cdot \mathfrak c (i)=i} - \prod_{j=1}^p (n_j)!, \end{equation} where $\mathfrak c$ is a $|\mathcal A|$-cycle permuting the groups corresponding to the elements $a\in\mathcal A$. Changing the order of summation in~\eqref{eq_x18}, we get $$ \sum_{q=1}^p\, \sum_{i=n_1+\dots+n_{q-1}+1}^{n_1+\dots+n_q} \, \sum_{\tilde g\in S_{n_1}\times S_{n_2}\times \dots \times S_{n_p}} \mathbf 1_{ \tilde g \cdot \mathfrak c (i)=i} - \prod_{j=1}^p (n_j)!. $$ In the internal sum over $\tilde g$, for the symmetric groups $S_{n_j}$ which permute the elements other than $ \mathfrak c (i)$ there are no restrictions and all $(n_j)!$ permutations are possible. However, if $\mathfrak c(i)$ belongs to the indices permuted by $S_{n_j}$, then there are two options: if~$\mathfrak c(i)=i$, then $(n_j-1)!$ permutations satisfy the restriction $ \tilde g \cdot \mathfrak c (i)=i$, otherwise no permutations satisfy the restriction. Hence, summing over $\tilde g$ and then $i$, we get $$\sum_{q=1}^p \left[\mathbf 1_{q\in \mathcal A} (n_q-1)+\mathbf 1_{q\not\in\mathcal A} n_q \right] (n_q-1)! \left[\prod_{j\ne q} (n_j)!\right] - \prod_{j=1}^p (n_j)!. $$ The last expression can be transformed into $$ (p-1)\prod_{j=1}^p (n_j)! - \sum_{q=1}^p \mathbf 1_{q\in \mathcal A} (n_q-1)! \left[\prod_{j\ne q} (n_j)!\right]. $$ Dividing by $\prod\limits_{j=1}^p (n_j)!$, we get~\eqref{eq_N11_formula}. \end{proof} \subsection{\texorpdfstring{$a\dots ab\dots b$}{a...ab...b} case with \texorpdfstring{$p=2$}{p=2}} Another situation where the general formula of Theorem~\ref{Theorem_Traces_general} is fully explicit is $p=2$. In fact, in this special case we are able to compute all eigenvalues $\mathrm{eig}_m(\lambda,\tau,i)$ rather than only their sums over $i$. We fix $N=1,2,\dots$ and $1\le \eta \le N-1$ and set the multiplicities to be $(n_1,n_2)= (N-\eta,\eta)$. We further choose $a>b\ge 0$ and set \begin{equation} \label{eq_two_row} \lambda= \underbrace{a,a,\dots,a}_{N-\eta}, \underbrace{b,b,\dots,b}_{\eta}. \end{equation} \begin{coro} \label{Corollary_ab} For $\lambda$ in~\eqref{eq_two_row} and $\tau=(N-k,k)$ with $0\le k\le \min(\eta,N-\eta)$, we have $$ \mathrm{eig}_m(\lambda,\tau,i) = (a+\theta \eta)^m (N-\eta) + b^m \eta-\theta \frac{(a+\theta \eta)^m- b^m}{a+\theta \eta -b} \bigl(\eta (N-\eta) -k(N-k+1)\bigr) , $$ for all $1\le i\le \dim V_{\lambda,\tau}$. For other $\tau$, the space $V_{\lambda,\tau}$ is empty. \end{coro} \begin{proof} We need some facts from the representation theory of symmetric groups, which can be found e.g.\ in~\cite{dunkl1978addition}, or \cite[Section 3]{Stanton1984}, or \cite[Sections 6.1, 6.2]{ceccherini2008harmonic}. In particular, $V_\lambda$ expands into a direct multiplicity one sum of irreducible representations $\tau=(N-k,k)$, $0\le k\le \min(\eta,N-\eta)$. The operator $(T_1^m+\dots+T_N^M)$ commutes with the action of the symmetric group. Therefore, by Schur's lemma, it necessarily acts as a multiple of the identity in each $V_{\lambda;\tau}$ (which would not have been true if $\tau$ had multiplicity larger than $1$ in $\lambda$). Hence, the eigenvalues $\mathrm{eig}_m(\lambda,\tau,i)$ do not depend on $i$ and we can get them by dividing~\eqref{eq_general_trace} by $\dim \tau$. Let us evaluate the ingredients of~\eqref{eq_general_trace}. $(S_N,S_{N-\eta}\times S_\eta)$ is a Gelfand pair, which means that each irreducible representation of $S_N$ has at most one $S_{N-\eta}\times S_\eta$-invariant vector. Interpreting summations over $\tilde g \in S_{N-\eta}\times S_\eta$ in~\eqref{eq_family_character} as projections on invariants, we identify $ \chi^\tau[\mathcal A; \mathbf n]$ with spherical functions corresponding to this invariant vector. The latter spherical function is evaluated in \cite[Corollary 2.2]{dunkl1978addition}, \cite[(3.15)]{Stanton1984}, \cite[Theorems 6.1.10, 6.2.3]{ceccherini2008harmonic} in terms of the hypergeometric function $_3F_2$, which can also be identified with the Hahn polynomial. The value at length $2$ cycle is given as \begin{equation} _3 F_2\left( \begin{smallmatrix}-k,\, k-N-1,\, -1 \\ \eta-N, -\eta \end{smallmatrix} \Bigr| 1 \right)=1 + \frac{(-k) (k-N-1) (-1)}{(-\eta)(\eta-N) 1!}=1-\frac{k(N-k+1)}{\eta(N-\eta)}. \end{equation} Hence,~\eqref{eq_general_trace} specializes to: \begin{multline*} \sum_{\begin{smallmatrix}\mathcal A\subset \{1,2\}\\ |\mathcal A|=1\end{smallmatrix}} h^{\mathcal A}_{m} \cdot \prod_{\alpha\in \mathcal A} n_\alpha -\theta \sum_{\begin{smallmatrix}\mathcal A\subset \{1,2\}\\ |\mathcal A|=2\end{smallmatrix}} h^{\mathcal A}_{m-1} \cdot \left(1-\frac{k}{\eta}\cdot \frac{N-\eta+1}{N-\eta }\right)\cdot \prod_{\alpha\in \mathcal A} n_\alpha. \\=(a+\theta \eta)^m (N-\eta) + b^m \eta-\theta \frac{(a+\theta \eta)^m- b^m}{a+\theta \eta -b} \left(1-\frac{k}{\eta}\cdot \frac{N-k+1}{N-\eta }\right) \eta (N-\eta). \end{multline*} \end{proof} \subsection{More complicated isotypes} In the follow-up work, the spherical functions $\chi\left[ \mathcal{A},\mathbf{n}\right] $ have been computed for the hook isotypes $\left[ N-b,1^{b}\right] $ for $p>b$ in \cite{Dun25a} and for the isotypes $\left[ N-k,k\right] $ for $p=3$ in \cite{Dun25b}. \section{Example: Eigenvalues for \texorpdfstring{$N=3$}{N=3}.} \label{Section_N3} Here is the full list of the eigenvalues of the operator $\PP_m$ for $N=3$. We keep using the notation~\eqref{eq_x3}. When we speak about leading monomials in the description of eigenfunctions, we use the point of view of Theorem~\ref{Theorem_classification_of_ef} and describe the part of $F_{\lambda,\tau,i}$ in~$V_\lambda$. \begin{enumerate}[label=(\roman*)] \item $\lambda=(a,a,a)$: For each $a\ge 0,$ $\PP_m$ has an eigenvalue $3 a^2 m$ on the eigenfunction $x_1^a x_2^a x_3^a$. \item $\lambda=(a,a,b)$: For each $a>b \ge 0$, $\PP_m$ has: \begin{enumerate} \item Eigenvalue $2 (a+\theta)^m + b^m - 2 \theta\frac{(a+\theta)^m-b^m}{a+\theta-b}$ on the eigenfunction with leading monomials $x_1^a x_2^a x_3^b + x_1^a x_2^b x_3^a+ x_1^b x_2^a x_3^a$. \item Two equal eigenvalues $2 (a+\theta)^m + b^m + \theta\frac{(a+\theta)^m-b^m}{a+\theta-b}$ on two eigenfunctions with leading monomials in the space spanned by $x_1^a x_2^a x_3^b - x_1^a x_2^b x_3^a$ and $x_1^a x_2^a x_3^b - x_1^b x_2^a x_3^a$. \end{enumerate} \item $\lambda=(a,b,b)$: For each $a>b \ge 0$, $\PP_m$ has: \begin{enumerate} \item Eigenvalue $(a+2\theta)^m + 2b^m - 2 \theta\frac{(a+2\theta)^m-b^m}{a+2\theta-b}$ on the eigenfunction with leading monomials $x_1^a x_2^b x_3^b + x_1^b x_2^a x_3^b+ x_1^b x_2^b x_3^a$. \item Two equal eigenvalues $(a+2\theta)^m + 2b^m + \theta\frac{(a+2\theta)^m-b^m}{a+2\theta-b}$ on two eigenfunctions with leading monomials in the space spanned by $x_1^a x_2^b x_3^b - x_1^b x_2^a x_3^b$ and $x_1^a x_2^b x_3^b - x_1^b x_2^b x_3^a$. \end{enumerate} \item $\lambda=(a,b,c)$: For each $a>b>c \ge 0$, $\PP_m$ has (we denote $(\ell_1,\ell_2,\ell_3)=(a+2\theta,b+\theta,c)$): \begin{enumerate} \item Eigenvalue $h^{(1)}_m\bigl(\ell_1,\ell_2,\ell_3\bigr)-\theta h^{(2)}_{m-1}\bigl(\ell_1,\ell_2,\ell_3\bigr)+\theta^2 h^{(3)}_{m-2}\bigl(\ell_1,\ell_2,\ell_3\bigr)$ on the eigenfunction with leading monomials $$x_1^a x_2^b x_3^c +x_1^b x_2^a x_3^c+x_1^a x_2^c x_3^b+x_1^c x_2^b x_3^a+x_1^b x_2^c x_3^a+x_1^c x_2^a x_3^b.$$ \item Two pairs of equal eigenvalues \begin{multline} \label{eq_irrational_ev} \left( \ell_{1}^{m}+\ell_{2}^{m}+\ell_{3}^{m}\right) -\frac{\theta^{2}}{2}h_{m-2}\left( \ell_{1},\ell_{2},\ell_{3}\right) \\ \pm\frac{\theta}{2} h_{m-2}\left( \ell_{1},\ell_{2},\ell_{3}\right) \sqrt{4\left( \ell_{1}^{2}+\ell_{2}^{2}+\ell_{3}^{2}\right) -4\left( \ell_{1}\ell_{2}+\ell_{1}\ell_{3}+\ell_{2}\ell_{3}\right) -3\theta^{2}} \end{multline} on four eigenfunctions with leading monomials in the space spanned by \begin{equation} \label{eq_x17} x_1^a x_2^b x_3^c-x_1^b x_2^c x_3^a,\quad x_1^a x_2^b x_3^c-x_1^c x_2^a x_3^b,\quad x_1^b x_2^a x_3^c-x_1^a x_2^c x_3^b,\quad x_1^b x_2^a x_3^c-x_1^c x_2^b x_3^a. \end{equation} \item Eigenvalue $h^{(1)}_m\bigl(\ell_1,\ell_2,\ell_3\bigr)+\theta h^{(2)}_{m-1}\bigl(\ell_1,\ell_2,\ell_3\bigr)+\theta^2 h^{(3)}_{m-2}\bigl(\ell_1,\ell_2,\ell_3\bigr)$ on the eigenfunction with leading monomials $$x_1^a x_2^b x_3^c -x_1^b x_2^a x_3^c-x_1^a x_2^c x_3^b-x_1^c x_2^b x_3^a+x_1^b x_2^c x_3^a+x_1^c x_2^a x_3^b.$$ \end{enumerate} \end{enumerate} (i.a), (ii.a), (iii.a), and (iv.a) are all instances of Theorem~\ref{Theorem_symmetric_ef} or Theorem~\ref{Theorem_Traces_general} with $\tau=(3)$. (ii.ab) and (iii.ab) are instances of Corollary~\ref{Corollary_ab}. (iv.a) and (iv.c) are instances of Theorem~\ref{Theorem_Traces_distinct} for $\tau=(3)$ and $\tau=(1,1,1)$, respectively. The sum of all four eigenvalues in (iv.b) matches Theorem~\ref{Theorem_Traces_distinct} for $\tau=(2,1)$, where we have $\dim\tau=\chi^{\tau}(\mathrm{Id})=2$, $\chi^{\tau}\bigl( (1,2)\bigr)=0$, $\chi^{\tau}\bigl( (1,2,3)\bigr)=-1$. The individual eigenvalues in~(iv.b) are not computed by any of our theorems; the formula~\eqref{eq_irrational_ev} can be obtained by evaluating the matrices of the operators $T_1^m$, $T_2^m$, $T_3^m$ by diagonalization and then direct computation of the eigenvalues of $T_1^m+T_2^m+T_3^m$ (we can either use $6\times 6$ matrices in the space of all permutations of $x_1^a x_2^b x_3^c$ or $4\times 4$ matrices in the subspace spanned by~\eqref{eq_x17}). \appendix \section{Basic properties of \texorpdfstring{$\PP_m$}{Pm}} \label{Section_appendix} In this section we discuss the basic properties of the operators $\PP_m$, mentioned in Section~\ref{Section_HP_operators}. These properties are well-known, we provide proofs only for the sake of being self-contained. \begin{lemma} \label{Lemma_commute} The operators $\PP_m$, $m=1,2,\dots$, mutually commute. \end{lemma} \begin{proof} We develop a table of commutators. We start by splitting the Dunkl operator of Definition~\ref{Definition_Dunkl} into two parts and write $\D_i=\frac{\partial}{\partial x_i} + \theta \Delta_i$ with $\Delta_i=\sum_{j\ne i} \frac{1-(i,j)}{x_i-x_j}$. We claim \begin{equation} \label{eq_Delta_commute} \Delta_a \Delta_b = \Delta_b \Delta_a,\qquad 1\le a,b \le N. \end{equation} In order to prove~\eqref{eq_Delta_commute} for $a\ne b$, we plug the definition of $\Delta_a$ into $\Delta_a\Delta_b$ and get $$ \sum_{j\ne a} \sum_{j'\ne b} \frac{1-(a,j)}{x_a-x_j} \cdot \frac{1-(b,{j'})}{x_b-x_{j'}}. $$ Note that the double sum splits into three types of summands: involving two, three, and four distinct variables $x_i$. The parts involving two and four variables are obviously the same in $\Delta_a \Delta_b$ and in $\Delta_b \Delta_a$. The part involving three variables, $x_a$, $x_b$, $x_j$ is more tricky, it is: \begin{equation} \label{eq_x9} \frac{1-(a,j)}{x_a-x_j} \cdot \frac{1-(b,j)}{x_b-x_j} + \frac{1-(a,b)}{x_a-x_b} \cdot \frac{1-(b,j)}{x_b-x_j} + \frac{1-(a,j)}{x_a-x_j} \cdot \frac{1-(b,a)}{x_b-x_a}, \end{equation} where in each fraction we first apply the operator in numerator and then divide by the denominator. The desired relation~\eqref{eq_Delta_commute} follows from the symmetry of~\eqref{eq_x9} in $a\leftrightarrow b$, which is verified directly by moving all permutation operators to the right. The next useful identity, which is verified directly, is that for $a\ne b$ \begin{equation} \label{eq_x14} [x_a,\Delta_b]=x_a \Delta_b- \Delta_b x_a=(a,b). \end{equation} Another direct verification yields that for $a\ne b$ we have \begin{equation} \label{eq_x15} \left[\frac{\partial}{\partial x_a},\Delta_b\right]=\frac{1-(a,b)}{(x_a-x_b)^2} +\frac{1}{x_a-x_b} \left(\frac{\partial}{\partial x_a}-\frac{\partial}{\partial x_b}\right) (a,b). \end{equation} The last identity makes it clear that $\left[\tfrac{\partial}{\partial x_a},\Delta_b\right]= \left[\tfrac{\partial}{\partial x_b},\Delta_a\right]$. Combining with~\eqref{eq_Delta_commute}, this implies commutativity of the Dunkl operators: $\D_a \D_b=\D_b \D_a$. Next, for each $ a \ne b $ we have \begin{equation} \label{eq_commutator} [x_a \D_a, x_b \D_b]= \theta (x_a \D_a - x_b \D_b) \cdot (a,b). \end{equation} Indeed, using~\eqref{eq_x14} and commutativity of $x_a$ with $\frac{\partial}{\partial x_b}$, we have $$ x_a \D_a x_b \D_b - x_b \D_b x_a \D_a= x_a x_b \D_a \D_b - x_a (a,b) \D_b - x_b x_a \D_b \D_a +\theta x_b (a,b) \D_a. $$ Cancelling the first and third terms and using $(a,b) \cdot \D_b \cdot (a,b)=\D_a$ we arrive at~\eqref{eq_commutator}. Using this identity we further compute \begin{multline} [x_a \D_a, (x_b \D_b)^m]=\sum_{k=0}^{m-1} (x_b \D_b)^k [x_a \D_a,x_b \D_b] (x_b\D_b)^{m-k-1}\\=\theta \sum_{k=0}^{m-1} (x_b \D_b)^k (x_a \D_a - x_b \D_b) \cdot (a,b) (x_b\D_b)^{m-k-1}= \\ \theta \sum_{k=0}^{m-1} (x_b \D_b)^k (x_a \D_a - x_b \D_b) \cdot (x_a\D_a)^{m-k-1} \cdot (a,b)= \theta \bigl((x_a \D_a)^m - (x_b \D_b)^m\bigr) \cdot (a,b). \end{multline} Iterating the same argument again, we get \begin{multline*} [(x_a \D_a)^l, (x_b \D_b)^m]=\sum_{k=0}^{l-1} (x_a \D_a)^k [x_a \D_a,(x_b \D_b)^m] (x_a\D_a)^{l-k-1}\\ =\theta \left( \sum_{k=m}^{m+l-1} (x_a \D_a)^{m+k} (x_b\D_b)^{m+l-k-1} - \sum_{k=0}^{l-1} (x_a \D_a)^k (x_b \D_b)^{m+l-k-1}\right) \cdot (a,b). \end{multline*} Interchanging $l$ and $m$ and using the skew-symmetry $[f,g]=-[g,f]$, we also have \begin{multline*} [(x_b \D_b)^l, (x_a \D_a)^m]\\ =\theta \left( \sum_{k=0}^{m-1} (x_a \D_a)^k (x_b \D_b)^{m+l-k-1}- \sum_{k=l}^{m+l-1} (x_a \D_a)^{l+k} (x_b\D_b)^{m+l-k-1}\right) \cdot (a,b). \end{multline*} Summing the last two identities over all $a