The invariant factors of the incidence matrices of points and subspaces in 𝑃𝐺(𝑛,𝑞) and 𝐴𝐺(𝑛,𝑞)

Abstract

We determine the Smith normal forms of the incidence matrices of points and projective ( r − 1 ) (r-1) -dimensional subspaces of PG ⁡ ( n , q ) \operatorname {PG}(n,q) and of the incidence matrices of points and r r -dimensional affine subspaces of AG ⁡ ( n , q ) \operatorname {AG}(n,q) for all n n , r r , and arbitrary prime power q q .