The definition of a generic initial ideal includes the assumption $x_1>x_2> \cdots >x_n$. A natural question is how generic initial ideals change when we permute the variables. In the article [1, §2], it is shown that the generic initial ideals are permuted in the same way when the variables in the monomial order are permuted. We give a different proof of this theorem. Along the way, we study the Zariski open sets which play an essential role in the definition of a generic initial ideal and also prove a result on how the Zariski open set changes after a permutation of the variables.
"A short note on generic initial ideals,"
Turkish Journal of Mathematics: Vol. 45:
3, Article 22.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss3/22