Families of Association Schemes on Triples from Two-Transitive Groups

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 $A\Gamma L(k,n)$, some subgroups of $P\Gamma L(k,n)$, and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of $P\Gamma L(k,n)$ and $A \Gamma L(k,n)$, and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.Comment: 20 pages, 5 table

Deterministic polynomial factoring over finite fields: A uniform approach via P-schemes

We introduce a family of combinatorial objects called P-schemes, where P is a collection of subgroups of a finite group G. A P-scheme is a collection of partitions of right coset spaces H\G, indexed by H ∈ P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as m-schemes (Ivanyos et al., 2009). We apply the theory of P-schemes to deterministic polynomial factoring over finite fields: suppose f(X) ∈ Z[X] and a prime number pare given, such that f(X) :=f(X) modpfactorizes into n =deg(f)distinct linear factors over the finite field F_p. We show that, assuming the generalized Riemann hypothesis (GRH), f(X)can be completely factorized in deterministic polynomial time if the Galois group G of f(X)is an almost simple primitive permutation group on the set of roots of f(X), and the socle of Gis a subgroup of Sym(k)for kup to 2^O(√log n). This is the first deterministic polynomial-time factoring algorithm for primitive Galois groups of superpolynomial order. We prove our result by developing a generic factoring algorithm and analyzing it using P-schemes. We also show that the main results achieved by known GRH-based deterministic polynomial factoring algorithms can be derived from our generic algorithm in a uniform way. Finally, we investigate the schemes conjecturein Ivanyos et al. (2009), and formulate analogous conjectures associated with various families of permutation groups. We show that these conjectures form a hierarchy of relaxations of the original schemes conjecture, and their positive resolutions would imply deterministic polynomial-time factoring algorithms for various families of Galois groups under GRH

Radicals of SnS_n-invariant positive semidefinite hermitian forms

Let G be a finite group, V a complex permutation module for G over a finite G-set X, and f:V 7V\u2192C a G-invariant positive semidefinite hermitian form on V. In this paper we show how to compute the radical V a5 of f, by extending to nontransitive actions the classical combinatorial methods from the theory of association schemes. We apply this machinery to obtain a result for standard Majorana representations of the symmetric groups

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.Comment: 23 page