More Combinatorics of Fulton’s Essential Set

Abstract We develop combinatorics of Fulton’s essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret Eriksson-Linusson’s characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of Bruhat-Ehresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.