ON THE DEGREES OF MINIMAL GENERATORS OF HOMOGENEOUS IDEALS IN THE EXTERIOR ALGEBRA

Abstract

We study how homogeneous ideals in the exterior algebra ∧ V over a finite-dimensional vector space V are minimally generated. In particular, we solve the following problems: Starting with an element p ν of degree ν, what is the maximum length ℓ of a sequence p υ,…,p υ+ℓ−1, with degpi = i, and such that pi is not in the ideal generated by p 1,…,p i−1 What is the maximal possible number of minimal generators of degree d of a homogeneous ideal which does not contain all elements of degree d + 1? Our main tool is the Kruskal–Katona theorem.