A generalization of Kruskal–Katona’s theorem

Let K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g_1, . . ., g_r such that deg g₁ ≤ deg g₂ ≤ · · · ≤ deg g_r. We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.