On the rank of incidence matrices in projective Hjelmslev spaces

Abstract

Let $$R$$ be a finite chain ring with $$|R|=q^m$$ , $$R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q$$ , and let $$\Omega ={{\mathrm{PHG}}}({}_RR^n)$$ . Let $$\tau =(\tau _1,\ldots ,\tau _n)$$ be an integer sequence satisfying $$m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0$$ . We consider the incidence matrix of all shape $$\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)$$ versus all shape $$\tau $$ subspaces of $$\Omega $$ with $$\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}$$ . We prove that the rank of $$M_{\varvec{m}^s,\tau }(\Omega )$$ over $$\mathbb {Q}$$ is equal to the number of shape $$\varvec{m}^s$$ subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all $$s$$ dimensional versus all $$t$$ dimensional subspaces of $${{\mathrm{PG}}}(n,q)$$ . We construct an example for shapes $$\sigma $$ and $$\tau $$ for which the rank of $$M_{\sigma ,\tau }(\Omega )$$ is not maximal.