Representations of generalized equipped posets and posets with automorphisms over Galois field extensions

Abstract

Generalized equipped posets and their representations over a Galois field extension K ⊂ L are introduced and studied. An equipment of a poset P is a pair ( Δ , S ) consisting of a finite group Δ and a collection S = { Δ x y | x ⩽ y } of its non-empty subsets such that Δ x y Δ y z ⊂ Δ x z for any chain x ⩽ y ⩽ z in P (equipped posets in the old sense of our former papers correspond to the case | Δ | = 2 with some additional restriction). Since the category of representations of a (strictly) equipped poset ( P , Δ , S ) is isomorphic to the category of representations of its evolvent ( Q , Δ ) (which is a poset with automorphisms), we study in fact representations of posets with automorphisms over the pair ( K , L ) . In the case Δ = Γ , where Γ = Gal ( L / K ) , we define the complexification functor C : rep ( K , L ) ( Q , Γ ) → rep L Q and the realification functor R : rep L Q → rep ( K , L ) ( Q , Γ ) and show that they induce reciprocal bijections between the isoclasses of indecomposables of the category rep ( K , L ) ( Q , Γ ) and the Γ-orbits of isoclasses of indecomposables of the category rep L Q . In this way, the problem on classification of indecomposables of equipped posets and posets with automorphisms over ( K , L ) actually is reduced to that one for representations of ordinary posets over L.