Research

Research Areas

Algebraic Combinatorics: Q-polynomial distance-regular graphs, association schemes, Terwilliger algebras, Leonard systems and tridiagonal pairs, algebraic graph theory

Algebra and Representation Theory: Double affine Hecke algebras, Onsager algebras, Askey–Wilson algebras, tridiagonal algebras, quantum algebras, orthogonal polynomials and special functions

Research Description

My research lies in algebraic combinatorics and its connections with representation theory and special functions. A central theme of my work is the algebraic structure of Q-polynomial distance-regular graphs and association schemes, particularly their connections with double affine Hecke algebras (DAHA) and related algebras. I study the interplay between the combinatorial and spectral structure of these objects and their associated algebraic and representation-theoretic structures, with particular interest in connections with orthogonal polynomials. Related areas of my research include Leonard systems and tridiagonal pairs, the Terwilliger algebra, the Onsager algebra, the Askey–Wilson algebra, and quantum algebras.

Publications

• On finite-dimensional multiplicity-free irreducible modules for a nil-DAHA of type (C₁ⱽ, C₁)
  J. Park, J.-H. Lee, and H. Baek
  Submitted. Preprint, 19pp (arXiv:2608.14299) 

• The nucleus of the Grassmann graph Jq(N,D)
  J.-H. Lee, J. Park, and I. Seong
  Discrete Math. 350 (2027), no. 1, Paper No. 115386, 20pp.  (arXiv:2509.15395)

• Every Q-polynomial distance-regular graph is sharp over  ℝ
  B. Fernandez, J.-H. Lee, and J. Park
  J. Combin. Theory Ser. A 224 (2026), Paper No. 106223, 36 pp. (arXiv:2511.19164)

• Four bases for the Onsager Lie algebra related by a ℤ₂×ℤ₂-action
  J.-H. Lee

J. Algebr. Comb. 63, no. 16 (2026) (arXiv: 2501.18364)

• 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
  Algebr. Comb. 8 (2025) no. 6, 1529--1546 (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. 

Honors and Awards

• NSF Travel Grant ($1600), Centre de Recherches Mathématiques (CRM) Workshop, Montréal, Canada, National Science Foundation, July 2022

• JSPS Postdoctoral Fellowship, Japan Society for the Promotion of Science, Japan, 2014-2016

• Excellence in Research Award, Department of Mathematics, University of Wisconsin-Madison, 2014