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Families of Association Schemes on Triples from Two-Transitive Groups

Jose Maria P. Balmaceda, Dom Vito A. Briones

Abstract

Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups by determining the orbits of the groups' two-point stabilizers. Specifically, we obtain these parameters for the ASTs obtained from the actions of $S_n$ and $A_n$, $PGU(3,q)$, $PSU(3,q)$, and $Sp(2k,2)$, $Sz(2^{2k+1})$ and $Ree(3^{2k+1})$, some subgroups of $AΓL(k,n)$, some subgroups of $PΓL(k,n)$, and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of $PΓL(k,n)$ and $A ΓL(k,n)$, and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.

Families of Association Schemes on Triples from Two-Transitive Groups

Abstract

Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups by determining the orbits of the groups' two-point stabilizers. Specifically, we obtain these parameters for the ASTs obtained from the actions of and , , , and , and , some subgroups of , some subgroups of , and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of and , and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.

Paper Structure

This paper contains 14 sections, 7 theorems, 16 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Association Schemes on Triples
  4. Two-Transitive Permutation Groups and ASTs
  5. ASTs from two-transitive groups
  6. Sk
  7. PGamL(k,n)
  8. PSL(2,n)
  9. AGamL(k,n)
  10. Sz(2 2k+1)
  11. Ree(3 2k+1)
  12. PGU(3,n)
  13. Sp(2k,2)
  14. Sporadics

Key Result

Theorem 1.1

The third valencies and number of relations of the ASTs obtained from the symmetric groups $S_k$, the projective semilinear groups $P\Gamma L(k,n)$, the projective special linear groups $PSL(2,n)$, the Suzuki groups $Sz(2^{2k+1})$, the Ree groups $Ree(3^{2k+1})$, the affine semilinear groups $A\Gamm

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Theorem 1.4, mesner_association_1990
  • Theorem 2.4: Corollary 2.8, mesner_association_1990
  • Lemma 2.5
  • Theorem 2.6: Theorem 4.1, mesner_association_1990
  • Remark 1
  • Remark 2
  • ...and 9 more