Invertible symmetric 3 × 3 binary matrices and GQ(2, 4)

Abstract

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 × 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2, 4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2, 2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2, 4) as a quadric in PG(5, 2) of projective index one. An interesting physics application of our findings is also mentioned.

Keywords:

• binary matrices of order 3 – GQ(2, 4) – PG(5, 2)
• quadratic forms; symplectic polarity

AMS Subject Classifications::

• 51E12
• 51E20

Acknowledgements

The work on this topic began in the framework of the ZiF Cooperation Group ‘Finite Projective Ring Geometries: An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics, and Chemistry of Coupling’, held in 2009 at the Center for Interdisciplinary Research (ZiF), University of Bielefeld, Germany. It was also partially supported by the VEGA grant agency, projects 2/0092/09 and 2/0098/10. We thank Prof. Hans Havlicek (Vienna) for several clarifying comments and Dr Petr Pracna (Prague) for an electronic version of the figure.