Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)

TL;DR

This paper uncovers a mathematical link between invertible symmetric 3x3 matrices over GF(2) and the structure of the generalized quadrangle GQ(2,4), revealing geometric and algebraic insights with potential physical applications.

Contribution

It establishes a detailed correspondence between matrices over GF(2) and points of GQ(2,4), including substructures and algebraic interpretations.

Findings

27 invertible symmetric 3x3 matrices over GF(2) relate to GQ(2,4) points.

15 matrices with eigenvalue one form a GQ(2,2) substructure.

12 matrices without eigenvalues correspond to a double-six configuration.

Abstract

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 x 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 physical application of our findings is also mentioned.