We define a set of invariants of a homogeneous ideal in a
polynomial ring called the symmetric iterated Betti numbers of .
We prove that
for
, the Stanley-Reisner ideal of a simplicial complex
, these numbers are the symmetric counterparts
of the exterior iterated Betti numbers of introduced by Duval
and Rose, and that the extremal Betti numbers of
are precisely the extremal (symmetric or exterior) iterated Betti numbers of
.
We show that the symmetric iterated Betti numbers of an ideal
coincide with those of a particular reverse lexicographic
generic initial ideal
Gin of , and interpret these
invariants in terms of the associated primes and standard pairs of
Gin. We close with results and conjectures about the
relationship between symmetric and exterior iterated Betti numbers
of a simplicial complex.