Presenting restricted Steinberg modules of general linear groups

We give generators and presentations for various local restrictions of Steinberg modules over fields and relate them to \emph{partial} Borel–Serre compactifications of locally symmetric spaces in the case of number fields, extending the existing theory of partial modular symbols for ${\mathrm{G}\mathrm{L}}_2(\text{Q})$. Along the way, we clarify the relationship between circuit'' and Bykovskii’'-type presentations for such modules. In an appendix, we relate the existence of such presentations to Koszulity properties of the Steinberg $\mathrm{V}\mathrm{B}$-algebra.