Symmetric iterated Betti numbers (Q598455)

The authors define, for any homogeneous ideal \(I\) in a polynomial ring, the symmetric iterated Betti numbers of \(I\). When \(I=I_{\Gamma}\) is the Stanley-Reisner ideal of a simplicial complex \(\Gamma\), the authors prove, using commutative algebra methods, that the symmetric iterated Betti numbers of \(I_{\Gamma}\) are the symmetric counterparts of the exterior iterated Betti numbers of \(\Gamma\) [see \textit{A. M. Duval} and \textit{L. L. Rose}, J. Algebr. Comb. 12, No.~3, 279--294 (2000; Zbl 0981.55010)] and that the exterior iterated Betti numbers of \(I_{\Gamma}\) are precisely the exterior iterated Betti numbers of \(\Gamma\). \noindent The authors give at the end of the paper some conjectures relating symmetric and exterior iterated Betti numbers.