A generalization of Kruskal–Katona’s theorem

Abstract

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 g1, . . ., gr such that deg g1 ≤ deg g2 ≤ · · · ≤ deg gr. 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.

Keywords

• exterior algebra
• hilbert functions
• monomial submodules
• lexicographic submodules
• primary 13a02, 15a75, 18g10
• secondary 68w30