Research

Research Areas

Algebraic combinatorics: Distance-regular graphs, Association schemes, Terwilliger algebra, Leonard pairs, Tridiagonal pairs, Algebraic graph theory 

Representation theory: Double affine Hecke algebra, Onsager algebra, Askey-Wilson algebra, Tridiagonal algebra, quantum algebra, Lie theory, Special functions. 

Research Description

My research interests focus on two mathematical objects: a combinatorial object known as a Q-polynomial distance-regular graph (DRG) and an algebraic object known as the double affine Hecke algebra (DAHA). Q-polynomial DRGs can be thought of as discrete analogues of rank-one symmetric spaces. The Q-polynomial property appears in many other areas of mathematics, such as Lie theory, quantum algebras, and orthogonal polynomials. The DAHA was introduced by Cherednik in the early 1990s. Since then, the theory of DAHA has garnered significant attention and has been connected to many other areas, such as integrable systems, algebraic geometry, quantum algebras, and orthogonal polynomials.  I discovered a relationship between Q-polynomial DRGs and the DAHA of rank one. This connection provides a new approach to the study of distance-regular graphs and DAHAs. Using this relationship, I explore connections between Q-polynomial DRGs and the non-symmetric (or Laurent) case of the terminating branch of the Askey scheme of orthogonal polynomials. Additionally, I have ongoing research into the generalized Terwilliger algebra and its connections to orthogonal polynomials, mathematical physics, quantum algebra, and Lie theory.

Publications

• Towards a classification of 1-homogeneous distance-regular graphs with positive intersection number a₁
  J.H. Koolen, M. Abdullah, B. Gebremichel, and J.-H. Lee
  Submitted, preprint, 19pp. (arXiv: 2404.01134) 

• The standard generators of the tetrahedron algebra and their look-alikes
  J.-H. Lee
  J. Algebra, 658 (2024) 556--591 (arXiv: 2405.05504)

• On the (non-)existence of tight distance-regular graphs: a local approach
  J.H. Koolen, J.-H. Lee, S. Li, Y. Li, X. Liang, and Y.-Y Tan
  Electron. J. Combin. 31(2) (2024), #P2.25 (arXiv: 2312.05595)

• A new feasibility condition for the AT4 family
  Z.-J. Xia, J.-H. Lee, and J.H. Koolen
  Electron. J. Combin. 30(2) (2023), #P2.7 (arXiv:2204.07842)

• Circular Hessenberg Pairs
  J.-H. Lee
  Linear Algebra Appl. 655 (2022), 201--235.  (arXiv:2209.02194)

• Remarks on pseudo-vertex-transitive graphs with small diameter
  J.H. Koolen, J.-H. Lee, and Y.-Y Tan
  Discrete Math. 345 (2022), no. 10, Paper No. 112990 (arXiv: 2102.00105)

• Multidimensional summability process on Baskakov-type approximation
  J.-H. Lee and R. Patterson
  Bull. Math. Anal. Appl. 14 (2022), no. 2, 12--23.

• Grassmann graphs, degenerate DAHA, and non-symmetric dual q-Hahn polynomials
  J.-H. Lee
  Linear Algebra Appl. 588 (2020), 160--195. (arXiv: 1809.08763) 

• Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual q-Krawtchouk polynomials
  J.-H. Lee and H. Tanaka
  SIGMA 14 (2018), 009, 27 pp. (arXiv:1709.07825)

• Dual polar graphs, a nil-DAHA of rank one, and non-symmetric dual q-Krawtchouk polynomials, Extended Abstract, Proceedings of FPSAC 2017
  J.-H. Lee and H. Tanaka
  Sém. Lothar. Combin. 78B #42 (2017), 12 pp.

• Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs
  J.-H. Lee
  J. Combin. Theory Ser. A, 147 (2017), 75--118. (arXiv:1509.04433)

• Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs
  J.-H. Lee
  RIMS Kôkyûroku, 1965 (2015), 101--111.

• Q-polynomial distance-regular graphs and a double affine Hecke algebra of rank one
  J.-H. Lee
  Linear Algebra Appl. 439 (2013), 3184--3240. (long version: arXiv:1307.5297, 79 pp.)

• A lower bound for the spectral radius of graphs with fixed diameter
  S.M. Cioaba, E.R. van Dam, J.H. Koolen, and J.-H. Lee
  European J. Combin. 31 (2010), 1560--1566.

• S.M. Cioaba, E.R. van Dam, J.H. Koolen, and J.-H. Lee
  Asymptotic results on the spectral radius and the diameter of graphs
  Linear Algebra Appl. 432 (2010), 722--737.