A NOTE ON GROUP INVARIANT INCIDENCE FUNCTIONS

Abstract

Partially ordered sets (X, ≼) and the corresponding incidence algebra I(X, F) are important algebraic structures also playing a crucial role for the enumeration, construction and the classification of many discrete structures. In this paper we consider partially ordered sets X on which some group G acts via the mapping X ×G → X, (x, g) 7→ xg and investigate such incidence functions ϕ : X × X → F of the incidence algebra I(X, F) which are invariant under the group action, i. e. which satisfy the condition ϕ(x, y) = ϕ(xg, yg) for all x, y ∈ X and g ∈ G. Within these considerations we define for such incidence functions ϕ the matrices ϕ∧ respectively ϕ∨ by summation of entries of ϕ and we investigate the structure of these matrices and generalize the results known from group actions on posets.