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Twisted Cherednik spectrum as a $q,t$-deformation

A. Mironov, A. Morozov, A. Popolitov

TL;DR

This work analyzes the twisted Cherednik integrable system with twist $a$ in the $N$-body representation, focusing on polynomial eigenfunctions at $t=q^{-m}$ and their deformation from the $q\to1$ limit. In the limit, eigenfunctions factor into a symmetric ground state $\omega_N^{(a,m)}$ and non-symmetric Jack-like excitations, and the authors develop a framework for a $q$-deformation that preserves a simple base structure: a $Q$-deformed ground state $\Omega_N^{(a,m)}$ and $a$-independent excitations described by non-symmetric Jack polynomials. They present explicit constructions for the ground state in low $N$ (notably $N=2$ and $N=3$) and several higher-$N$ cases, including $a=2$ and general $a>2$, using recursive, symmetric-polynomial expansions and direct $q$-deformations, with particular attention to the Schur- and Baker–Akhiezer-type structures. The results illustrate a rich, NP-like structure: while the eigenvalue equations are hard to solve, the ground state and low-lying excitations can be verified and constructed systematically; the work emphasizes the need for a rigorous deformation framework and potential creation-operator algebras in the DIM/baker–akhiezer context. Overall, the findings advance understanding of $q,t$-deformations in twisted Cherednik systems and connect to DIM algebra representations and symmetric polynomials in multiple variables.

Abstract

The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q\longrightarrow 1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\neq 1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.

Twisted Cherednik spectrum as a $q,t$-deformation

TL;DR

This work analyzes the twisted Cherednik integrable system with twist in the -body representation, focusing on polynomial eigenfunctions at and their deformation from the limit. In the limit, eigenfunctions factor into a symmetric ground state and non-symmetric Jack-like excitations, and the authors develop a framework for a -deformation that preserves a simple base structure: a -deformed ground state and -independent excitations described by non-symmetric Jack polynomials. They present explicit constructions for the ground state in low (notably and ) and several higher- cases, including and general , using recursive, symmetric-polynomial expansions and direct -deformations, with particular attention to the Schur- and Baker–Akhiezer-type structures. The results illustrate a rich, NP-like structure: while the eigenvalue equations are hard to solve, the ground state and low-lying excitations can be verified and constructed systematically; the work emphasizes the need for a rigorous deformation framework and potential creation-operator algebras in the DIM/baker–akhiezer context. Overall, the findings advance understanding of -deformations in twisted Cherednik systems and connect to DIM algebra representations and symmetric polynomials in multiple variables.

Abstract

The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of . Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at , which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.
Paper Structure (21 sections, 127 equations, 1 figure)

This paper contains 21 sections, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General description
  3. Twisted Cherednik system
  4. The $q\to 1$ limit
  5. On the principles and the base of $q$-deformation
  6. Expansion (\ref{['amundeformed']}) gets clear from the following example.
  7. A simple example: deformation at $N=2$
  8. The ground state $\Omega^{(2,m)}_N$
  9. $N=2$
  10. $N=3$
  11. $N>3$
  12. The ground state $\Omega^{(a,m)}_N$
  13. $N=3$, $m=1$ at any $a$
  14. $m=1,2$ at $a=3$
  15. Examples of $\Omega_2^{(2,1|3,m)}$.
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The spectrum of common polynomial eigenfunctions. Plotted on the vertical axis are the degrees of polynomials in $X_i$. There is a single ground state eigenfunction $\omega_N^{(a,m)}$ of degree $\frac{N(N-1)}{2}(\alpha-1)m$. Over it, there are the "excitations" of additional degrees $L$. They are labeled by weak compositions, so that there are $n_L$ different polynomials of the additional degree $L$. For particular values of $m$, there can be degeneracies, and some of these polynomials coincide, as well as their eigenvalue sets. Eigenvalues are not shown.