Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)
Algebraic Geometry
2012-10-01 v1 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.1009.1768,
title = {Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)},
author = {Andrea Blunck and Peter Levay and Metod Saniga and Peter Vrana},
journal= {arXiv preprint arXiv:1009.1768},
year = {2012}
}
Comments
10 pages, 1 figure