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

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={Δxy|x⩽y} of its non-empty subsets such that ΔxyΔyz⊂Δxz 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 and the realification functor 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 . 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.