Simplicial orders and chordality

Abstract

Chordal clutters in the sense of Bigdeli et al. (J Comb Theory Ser A 145:129---149, 2017) and Morales et al. (Ann Fac Sci Toulouse Ser 6 23(4):877---891, 2014) are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the $$\lambda $$ź-sequence of the chordal clutter. All possible $$\lambda $$ź-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the $$\lambda $$ź-sequence of a chordal clutter, we determine other numerical invariants of the circuit ideal, such as the $$\mathbf h $$h-vector and the Betti numbers.