Departure of Nicolas Crampé

Arriving on January 1, 2025, Nicolas Crampé left the CRM-CNRS (IRL3457) on December 31, 2025.

He worked with Luc Vinet, a professor at Université de Montréal at the Laboratoire de physique mathématique. He also had the opportunity to collaborate with Quentin Labriet, a post-doctoral researcher, and Lucia Morey, a post-doctoral researcher from Luc Vinet’s team. Pierre-Antoine Bernard, with whom Nicolas Crampé has worked since his master’s internship, defended his doctorate in June 2025, and they continued their collaboration.

Some of Nicolas Crampé’s collaborators visited him:

  • Gilles Parez (LAPTh, Annecy, France)
  • Loic Poulain d’Andecy (Université de Reims, France)
  • Wolter Groenevelt (University of Delft, Netherlands)
  • Rafael Nepomechie (University of Miami, USA)
  • Satoshi Tsujimoto (University of Kyoto, Japan)
  • Carel Wagenaar (University of Delft, Netherlands)
  • Meri Zaimi (Perimeter Institute, Waterloo, Canada).

Nicolas Crampé’s research mainly focuses on the relationships between representation theory, graph theory, special functions, and exactly solvable physical models. These links are particularly well established for Askey-Wilson polynomials. One of the objectives of his stay was to generalize these relationships to other functions.

For example, there are rational functions of Wilson that generalize these polynomials. In article [6], Nicolas Crampé defined algebras associated with these functions.

Another research direction involves studying multivariate polynomials. In articles [2] and [10], he introduced and studied algebras associated with bivariate Jacobi polynomials. The study of multivariate polynomials also naturally led him to classify the contiguity relations of univariate polynomials [4].

These relations allowed him to obtain interesting results for physical models or in representation theory, perfectly illustrating the interactions between these fields. Indeed, the contiguity relations enabled him to discover new exactly solvable physical models, such as XX [1], XY [3], and SSH [9] models. On the other hand, these same relations led him to introduce new Hopf structures for Lie algebras, thus allowing the definition of new quasi-Hopf algebras [8].

One of the important properties of physical models constructed from Askey-Wilson polynomials is the ability to design systems possessing the perfect transfer property. This property could be used in the design of quantum computers to transfer qubits faithfully. By studying some of these polynomials, Nicolas Crampé obtained new models possessing this property [7].

Finally, he also finalized a project concerning change of basis for tridiagonal pairs [5]. These pairs generalize Leonard pairs, for which the change of basis involves Askey-Wilson polynomials. This approach represents a promising new direction for defining new special functions.

Publications

During his stay, Nicolas Crampé wrote 10 research articles: 6 have been accepted in international research journals and 4 are submitted:

  1. su(2) symmetry of XX spin chains Nicolas Crampé, Rafael I. Nepomechie, Luc Vinet, Nabi Zare Harofteh Mathematical Physics, Analysis and Geometry and arXiv:2508.20184

  2. The rank two Jacobi algebra Nicolas Crampé, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov Annales Henri Poincaré and arXiv:2507.07766

  3. Exactly solvable inhomogeneous XY spin chain Pierre-Antoine Bernard, Nicolas Crampé, Quentin Labriet, Lucia Morey, Luc Vinet J Stat Mech and arXiv:2507.06331

  4. Contiguity relations for finite families of orthogonal polynomials in the Askey scheme Nicolas Crampé, Lucia Morey, Luc Vinet, Meri Zaimi The Ramanujan Journal 68 (2025) 68 and arXiv:2504.20802

  5. Change of basis for the tridiagonal pairs of type II Nicolas Crampé, Julien Gaboriaud, Satoshi Tsujimoto Nuclear Physics B 1018 (2025) 117083 and arXiv:2503.01231

  6. Algebras behind the bispectrality of the Wilson rational functions and their 4ϕ3 limits Nicolas Crampé, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov Numerical Algorithms and arXiv:2502.04275

  7. Perfect state transfer in inhomogeneous XX model of q-Racah type Nicolas Crampé, Simon Lafrance, Charles Robillard, Luc Vinet arXiv:2512.12419

  8. Polynomials of the Askey scheme as Clebsch–Gordan coefficients Nicolas Crampé, Loic Poulain d’Andecy, Luc Vinet arXiv:2511.21433

  9. Inhomogeneous SSH models and the doubling of orthogonal polynomials Nicolas Crampé, Quentin Labriet, Lucia Morey, Gilles Parez, Luc Vinet arXiv:2511.15527

  10. Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way arXiv:2509.07949

Nicolas Crampé is a research officer at CNRS.