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

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

2010-09-09

arxiv.org/abs/1009.1768v1

Mathematics

Algebraic Geometry

10 pages, 1 figure

Scientific paper

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.

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