BASES IN SYSTEMS OF SIMPLICES AND CHAMBERS

TATIANA ALEKSEYEVSKAYA

Abstract. We consider a finite set E of points in the n-dimensional affine space and two sets of objects that are generated by the set E: the system Σ of ndimensional simplices with vertices in E and the system Γ of chambers. The incidence matrix A =∥ a_{σ,γ} ∥, σ ∈ Σ, γ ∈ Γ, induces the notion of linear independence among simplices (and among chambers). We present an algorithm of construction of bases of simplices (and bases of chambers). For the case n = 2 such an algorithm was described in the author’s paper Combinatorial bases in systems of simplices and chambers (Discrete Mathematics 157 (1996) 15–37). However, the case of n-dimensional space required a different technique. It is also proved that the constructed bases of simplices are geometrical.