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Association Scheme on Triples from the Unitary Group

Jose Maria P. Balmaceda, Dom Vito A. Briones, Joris Buloron

TL;DR

The paper determines the intersection numbers of the association scheme on triples arising from the 2-transitive action of the finite unitary group on isotropic lines. It uses explicit geometric descriptions of the relations, orbit analysis on restricted sets, and fixed-field considerations to derive the full p_{ijk}^ℓ table, showing that a minimal 1D ternary subalgebra is generated by the adjacency hypermatrix A_4. This work completes the parametric understanding of ASTs for a major family of 2-transitive groups and highlights the role of a ternary algebra structure in encoding the intersection constants. The results integrate unitary-group geometry with higher-dimensional algebraic combinatorics, offering concrete tools for analyzing ASTs and their subalgebra structure.

Abstract

An association scheme on triples (AST) is a three-dimensional analogue of a classical association scheme. Similar to how a transitive group action produces a Schurian classical association scheme, a two-transitive group action produces an AST. This paper describes the relations and intersection numbers of ASTs from the finite unitary groups acting on the respective isotropic lines.

Association Scheme on Triples from the Unitary Group

TL;DR

The paper determines the intersection numbers of the association scheme on triples arising from the 2-transitive action of the finite unitary group on isotropic lines. It uses explicit geometric descriptions of the relations, orbit analysis on restricted sets, and fixed-field considerations to derive the full p_{ijk}^ℓ table, showing that a minimal 1D ternary subalgebra is generated by the adjacency hypermatrix A_4. This work completes the parametric understanding of ASTs for a major family of 2-transitive groups and highlights the role of a ternary algebra structure in encoding the intersection constants. The results integrate unitary-group geometry with higher-dimensional algebraic combinatorics, offering concrete tools for analyzing ASTs and their subalgebra structure.

Abstract

An association scheme on triples (AST) is a three-dimensional analogue of a classical association scheme. Similar to how a transitive group action produces a Schurian classical association scheme, a two-transitive group action produces an AST. This paper describes the relations and intersection numbers of ASTs from the finite unitary groups acting on the respective isotropic lines.

Paper Structure

This paper contains 7 sections, 8 theorems, 26 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Association Schemes on Triples
  4. Algebra of Hypermatrices
  5. ASTs from Two-Transitive Groups
  6. AST from the Unitary Group
  7. Parameters of the AST from the Unitary Group

Key Result

Theorem 2.1

Let $X=\{R_i\}_{i=0}^m$ be an AST with corresponding adjacency hypermatrices $\{A_i\}_{i=0}^m$. Then $\operatorname{Span}_\mathbb{C}(\{A_i\}_{i=0}^m)$ is a ternary algebra satisfying $A_{i}A_{j}A_{k}=\sum^{m}_{\ell=0}{p_{ijk}^{\ell}A_{\ell}},$ for any $i,j,k\in\left\{0,1,\ldots,m\right\}$. Moreover,

Theorems & Definitions (17)

  • Definition 2.1
  • Remark 2.1: Proposition $2.7$, mb
  • Theorem 2.1: Theorem $1.4$ and Corollary 2.8, mb
  • Theorem 2.2: Theorem $4.1$, mb
  • proof : Outline of proof.
  • Remark 2.2: Remark 2.10 bb, Lemma 4.2 mb
  • Theorem 3.1: Table $1$, bb
  • Proposition 3.1
  • proof
  • Corollary 3.1: Table $1$, bb
  • ...and 7 more