The symplectic trilinear mappings; an algorithmic approach of the classification; case of the field GF(3)

Abstract

Let V and W be vector spaces over a commutative field K with n=dimension(V). The set AT(n,m,K) of the skew-symmetric trilinear mappings from V³ to W whose images has rank m is provided with natural actions of both linear groups GL(V) and GL(W). We consider the problem of counting orbits. For m=1,2,3, the number a(n,m) of orbits of elements of AT(n,m,GF(3)) is shown to coincide with the total number STH(n,m) of pairwise non-isomorphic Hall systems whose rank and 3-order are respectively n+1 and n+m. For m>3, STH(n,m) is at least 3+a(n,m). A computational approach was used to obtain a set of representatives of AT(5,2,GF(3)), and correspondingly an exhaustive list of the order 3⁷ Hall systems: there are 13 such systems. Two algorithms were used: one proceeding to random changes of basis for partial determination of orbits, and another one computing invariants in order to show that some elements are not related.