Generic Initial Ideals Lecture 5

Let ∆ be a simplicial complex on [n]. Let 1 ≤ i < j ≤ n. Write
Shift ij (∆) for the collection of subsets of [n] consisting of the
sets C ij (F) ⊂ [n], where F ∈ ∆ and where
{ (F \ {i}) ∪ {j}, if i ∈ F, j ∉ F and (F \ {i}) ∪ {j} ∉ ∆,
C ij (F) =
F, otherwise.
Proposition 1: (a) Shift ij (∆) is a simplicial complex on [n], and
the operation ∆ → Shift ij (∆) satisfies the conditions (S 2 ), (S 3 )
and (S 4 ).