We define a set of invariants of a homogeneous ideal
![$ I$](img22.png)
in a
polynomial ring called the symmetric iterated Betti numbers of
![$ I$](img22.png)
.
We prove that
for
![$ I_{\Gamma}$](img24.png)
, the Stanley-Reisner ideal of a simplicial complex
![$ \Gamma$](img25.png)
, these numbers are the symmetric counterparts
of the exterior iterated Betti numbers of
![$ \Gamma$](img25.png)
introduced by Duval
and Rose, and that the extremal Betti numbers of
![$ I_\Gamma$](img26.png)
are precisely the extremal (symmetric or exterior) iterated Betti numbers of
![$ \Gamma$](img25.png)
.
We show that the symmetric iterated Betti numbers of an ideal
![$ I$](img22.png)
coincide with those of a particular reverse lexicographic
generic initial ideal
Gin
![$ \,(I)$](img27.png)
of
![$ I$](img22.png)
, and interpret these
invariants in terms of the associated primes and standard pairs of
Gin
![$ \,(I)$](img27.png)
. We close with results and conjectures about the
relationship between symmetric and exterior iterated Betti numbers
of a simplicial complex.