Incidence Gain Graphs and Generalized Quadrangles
Ryan McCulloch
TL;DR
The paper develops a construction of generalized quadrangles from incidence geometries by decorating the incidence graph $\Gamma$ with a gain group acting on a set $\Lambda$. The resulting incidence structure $\mathfrak{M}(\Gamma,\varphi)$ is a generalized quadrangle if and only if the rho maps $\rho_{b,p,\lambda}$ are bijective for all $p\in\mathscr{P}$, $b\in\mathscr{B}$ with $p\cancel{\mathrm{I}} b$, and $\lambda\in\Lambda$, in which case $X$ forms an ovoid and $(\mathscr{P},\mathscr{B},\mathrm{I})$ is a Steiner system with parameters yielding the GQ incidence relations. The affine-plane example over a field $\mathbb{F}$ yields a concrete family of generalized quadrangles, and for finite $|\mathbb{F}|=q$ the constructed GQ is isomorphic to the dual Payne quadrangle $P(W(q),x)$. Structural consequences include explicit ovoids and Steiner-system incidence, with open questions on classification of gain functions on affine planes and potential isomorphism questions across different gains.
Abstract
We demonstrate a construction method based on a gain function that is defined on the incidence graph of an incidence geometry. Restricting to when the incidence geometry is a linear space, we show that the construction yields a generalized quadrangle provided that the gain function satisfies a certain bijective property. Our method is valid for finite and infinite geometries. We produce a family of generalized quadrangles by defining such a gain function on an affine plane over an arbitrary field.