Multivariate P- and Q-polynomial structures of generalized Johnson association schemes

@misc{kurihara2026multivariate,
  title = {{Multivariate $P$- and $Q$-polynomial structures of generalized Johnson association schemes}},
  author = {Hirotake Kurihara},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22058},
  eprint = {2610.22058},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22058},
}

TY  - UNPB
TI  - Multivariate P- and Q-polynomial structures of generalized Johnson association schemes
AU  - Hirotake Kurihara
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22058
UR  - https://arxiv.org/abs/2610.22058
AB  - Let X be a commutative association scheme of class d on X. We construct the generalized Johnson association scheme J_n,h(X) on partial maps from [n] to X with domains of size h. For 1≤ h≤ n−1, we prove that J_n,h(X) has (d+1)-variate P- and Q-polynomial structures. In particular, its multivariate Q-polynomial property answers affirmatively a problem posed for generalized Johnson association schemes arising from finite Gelfand pairs. We also determine the primitive idempotents and the two eigenmatrices, expressing the first eigenmatrix as a product of an Eberlein polynomial and a multivariate Krawtchouk polynomial, and the second eigenmatrix as a product of a Hahn polynomial and a multivariate Krawtchouk polynomial. For the multivariate factors, we use Mizukawa--Tanaka (d+1,2d+2)-hypergeometric functions. In an appendix, we identify the normalized columns of the second eigenmatrix with the known spherical functions in the finite Gelfand pair case. We also discuss the orthogonality relations and the imprimitivity arising from partial maps with the same domain, and explicitly describe the multivariate P- and Q-polynomial structures inherited by the blocks and the quotient.
ER  -