Families of Association Schemes on Triples from Two-Transitive Groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916283719614464 |
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| author | Balmaceda, Jose Maria P. Briones, Dom Vito A. |
| author_facet | Balmaceda, Jose Maria P. Briones, Dom Vito A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_07753 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Families of Association Schemes on Triples from Two-Transitive Groups Balmaceda, Jose Maria P. Briones, Dom Vito A. Combinatorics Group Theory 05E30, 20B20 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. |
| title | Families of Association Schemes on Triples from Two-Transitive Groups |
| topic | Combinatorics Group Theory 05E30, 20B20 |
| url | https://arxiv.org/abs/2107.07753 |