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 degp_i = i, and such that p_i 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.

ACKNOWLEDGMENTS

Snellman was supported by a grant from Svenska Institutet and by grant no 231801F from Centre International des Étudiants et Stagiaires while visiting Laboratoire GAGE.