A GEOMETRIC APPROACH TO POLYNOMIAL PROPERTY FOR PATTERN SUBGROUPS IN ${{\mathrm {\mathbf {U}}}}_{\boldsymbol {a,b,c,d}}$

Abstract Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$ , where $a+b+c+d=n$ . In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$ . We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.