[Paper Review] Bivariate $P$-polynomial association schemes

Pierre-Antoine Bernard, Nicolas Crampé+3 more|arXiv (Cornell University)|Dec 21, 2022

Coding theory and cryptography28 references4 citations

TL;DR

This paper introduces bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$, generalizing classical $P$-polynomial schemes by replacing univariate polynomials with bivariate ones. It establishes an equivalence between these schemes and specific intersection number constraints, demonstrating that known higher-rank schemes—including direct products, symmetrized schemes, the 24-cell, non-binary Johnson schemes, and those from isotropic/attenuated spaces—fall into this class, with analogous $Q$-polynomial variants also defined.

ABSTRACT

Bivariate P-polynomial association scheme of type $(α,β)$ are defined as a generalization of the P-polynomial association schemes. This generalization is shown to be equivalent to a set of conditions on the intersection parameters. A number of known higher rank association schemes are seen to belong to this broad class. Bivariate Q-polynomial association schemes are similarly defined.

Motivation & Objective

• To generalize the concept of $P$-polynomial association schemes from univariate to bivariate polynomials.
• To define and characterize bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$ using a novel partial order on monomials.
• To establish an equivalence between the bivariate $P$-polynomial property and specific constraints on intersection numbers.
• To extend the framework to $Q$-polynomial schemes and demonstrate their presence in known association schemes.
• To explore connections with orthogonal polynomials, subconstituent algebras, and potential multivariate generalizations.

Proposed method

• Define a total order deg-lex on monomials and introduce a partial order $\preceq_{(\alpha,\beta)}$ for bivariate polynomials with parameters $0 \leq \alpha \leq 1$, $0 \leq \beta < 1$.
• Introduce the notion of bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$, where adjacency matrices satisfy $A_i = v_i(A_1, A_2)$ for bivariate polynomials $v_i$ of degree $i$ under the $\preceq_{(\alpha,\beta)}$ order.
• Derive recurrence relations for the bivariate polynomials $v_{ij}$ using intersection number constraints, leading to a system of linear recurrences.
• Construct the idempotents of the Bose–Mesner algebra using the eigenvalues $\theta_{ij}$ and $\mu_{ij}$ of the adjacency matrices $A_{10}$ and $A_{01}$, derived from the scheme parameters.
• Define bivariate $Q$-polynomial schemes analogously, using dual eigenvalues and verifying the property via symmetrization and direct product constructions.
• Establish a Wilson-type duality relation between the polynomials $v_{ij}$ and their duals $v^\star_{mn}$, linking the $P$- and $Q$-polynomial structures.

Experimental results

Research questions

• RQ1 How can the classical notion of $P$-polynomial association schemes be generalized to bivariate polynomials while preserving structural coherence?
• RQ2 What constraints on intersection numbers characterize bivariate $P$-polynomial schemes of type $(\alpha,\beta)$?
• RQ3 Which known higher-rank association schemes, such as the 24-cell or non-binary Johnson scheme, satisfy the bivariate $P$-polynomial property?
• RQ4 Can the $Q$-polynomial property be similarly generalized to bivariate settings, and which schemes exhibit this?
• RQ5 What algebraic structures, such as subconstituent algebras or generalized Leonard pairs, emerge from bivariate $P$- and $Q$-polynomial schemes?

Key findings

• The bivariate $P$-polynomial association scheme of type $(\alpha,\beta)$ is equivalent to a system of recurrence relations for the polynomials $v_{ij}$, involving parameters $k, b, c, \theta, \tau$.
• The direct product of $P$-polynomial schemes and symmetrized schemes yield bivariate $P$-polynomial schemes, as do the 24-cell, non-binary Johnson scheme, and schemes from isotropic or attenuated spaces.
• The symmetrization of a $P$-polynomial scheme produces a bivariate $P$- and $Q$-polynomial scheme, with explicit duality relations between the polynomials $v_{ij}$ and $v^\star_{mn}$.
• The eigenvalues $\theta_{ij}$ and $\mu_{ij}$ of $A_{10}$ and $A_{01}$ are given by $\theta_{ij} = (N-i-j)k + i\theta + j\tau$ and $\mu_{ij} = (N-i-j)\frac{kb}{c} - i(\theta+1) - j(\tau+1)$, respectively.
• A Wilson-type duality holds: $\frac{i!j!(N-i-j)!}{k^i (bk/c)^j} v_{ij}(\theta_{mn}, \mu_{mn}) = \frac{m!n!(N-m-n)!}{(k^\star)^m (b^\star k^\star / c^\star)^n} v^\star_{mn}(\theta^\star_{ij}, \mu^\star_{ij})$, confirming the duality.
• The subconstituent algebra of such schemes is generated by four elements, suggesting a generalization of tridiagonal relations and Leonard pairs to higher-rank settings.