A translation generalized quadrangle in characteristic $\ne 0$ is linear

TL;DR

This paper proves that all translation generalized quadrangles in positive characteristic are linear, confirming a long-standing conjecture by embedding them into linear structures and establishing a notion of characteristic.

Contribution

It demonstrates that any translation generalized quadrangle can be embedded into a linear one and confirms linearity in positive characteristic, advancing understanding of their structure.

Findings

Any translation generalized quadrangle can be embedded in a linear one.

In positive characteristic, all translation generalized quadrangles are linear.

Introduces a notion of characteristic for these structures.

Abstract

It is a long-standing conjecture from the 1970s that every translation generalized quadrangle is linear, that is, has an endomorphism ring which is a division ring (or, in geometric terms, that has a projective representation). We show that any translation generalized quadrangle $\Gamma$ is ideally embedded in a translation quadrangle which is linear. This allows us to weakly represent any such $\Gamma$ in projective space, and moreover, to have a well-defined notion of "characteristic" for these objects. We then show that each translation quadrangle in positive characteristic indeed is linear.