Trivariate P- and Q-polynomial structures for interior shells of the Grassmann scheme
Abstract
Association schemes induced on the shells of the Grassmann scheme Grq(N,a) form an important family in the study of multivariate P- and Q-polynomial structures. For a<N/2, the scheme induced on the furthest shell Xa is the known bivariate P- and Q-polynomial scheme arising from attenuated spaces. In this paper, we prove that, for every 1≤ i≤ a−1, the association scheme induced on the interior shell Xi is both trivariate P-polynomial and trivariate Q-polynomial. The two structures share a common index set consisting of the lattice points of a bounded non-simplicial polyhedron. We establish the P-polynomial property through combinatorial interpretations of the intersection numbers. Using Dunkl's coordinate description of Xi and the addition theorem, together with recurrence relations for the q-Hahn and q-Krawtchouk polynomials, we obtain the required vanishing and nonvanishing conditions for the Krein parameters, and hence the Q-polynomial structure. Motivated by this index set, we extend the notion of an AM-Leonard pair from simplicial to polyhedral domains and show that every interior shell carries a polyhedral A3-Leonard pair.
Related
- Multivariate P- and Q-polynomial structures of generalized Johnson association schemes
- Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers
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- The Classification of the Rota-Baxter Operators on Four-dimensional Grassmann Algebra
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BibTeX
@misc{zhu2026trivariate,
title = {{Trivariate $P$- and $Q$-polynomial structures for interior shells of the Grassmann scheme}},
author = {Yan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19974},
eprint = {2610.19974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19974},
}
RIS
TY - UNPB
TI - Trivariate P- and Q-polynomial structures for interior shells of the Grassmann scheme
AU - Yan Zhu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19974
UR - https://arxiv.org/abs/2610.19974
AB - Association schemes induced on the shells of the Grassmann scheme Gr_q(N,a) form an important family in the study of multivariate P- and Q-polynomial structures. For a<N/2, the scheme induced on the furthest shell X_a is the known bivariate P- and Q-polynomial scheme arising from attenuated spaces. In this paper, we prove that, for every 1≤ i≤ a−1, the association scheme induced on the interior shell X_i is both trivariate P-polynomial and trivariate Q-polynomial. The two structures share a common index set consisting of the lattice points of a bounded non-simplicial polyhedron. We establish the P-polynomial property through combinatorial interpretations of the intersection numbers. Using Dunkl's coordinate description of X_i and the addition theorem, together with recurrence relations for the q-Hahn and q-Krawtchouk polynomials, we obtain the required vanishing and nonvanishing conditions for the Krein parameters, and hence the Q-polynomial structure. Motivated by this index set, we extend the notion of an A_M-Leonard pair from simplicial to polyhedral domains and show that every interior shell carries a polyhedral A_3-Leonard pair.
ER -