A NOTE ON GROUP INVARIANT INCIDENCE FUNCTIONS

Authors : Michael BRAUN

Pages : 90-99

View : 14 | Download : 9

Publication Date : 2007-12-01

Article Type : Research Paper

Abstract :Partially ordered sets insert ignore into journalissuearticles values(X, ≼); and the corresponding incidence algebra Iinsert ignore into journalissuearticles values(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, insert ignore into journalissuearticles values(x, g); 7→ xg and investigate such incidence functions ϕ : X × X → F of the incidence algebra Iinsert ignore into journalissuearticles values(X, F); which are invariant under the group action, i. e. which satisfy the condition ϕinsert ignore into journalissuearticles values(x, y); = ϕinsert ignore into journalissuearticles values(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.
Keywords : Incidence algebra, group action, Plesken matrices