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

  • L. Beneteau
Submitted Contributions Computational Algebra And Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 508)


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.


Isomorphy Class Affine Space Moufang Loop Transfer Theorem Commutative Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • L. Beneteau
    • 1
  1. 1.INSA, MathématiqueToulouse CedexFrance

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