The affinity of a permutation of a finite vector space

Abstract

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k -affinity ( f ) denote the number of k-flats X of V such that f ( X ) is also a k-flat. By k -spectrum ( n , q ) we mean the set of integers k -affinity ( f ) , where f runs through all permutations of V. The problem of the complete determination of k -spectrum ( n , q ) seems very difficult except for small or special values of the parameters. However, we are able to determine ( n - 1 ) -spectrum ( n , 2 ) and establish that 0 ∈ k -spectrum ( n , q ) in the following cases: (i) q ≥ 3 and 1 ≤ k ≤ n - 1 ; (ii) q = 2 , 3 ≤ k ≤ n - 1 ; (iii) q = 2 , k = 2 , n ≥ 3 odd. For 1 ≤ k ≤ n - 1 and ( q , k ) ≠ ( 2 , 1 ) , the maximum of k -affinity ( f ) is obtained when f is any semiaffine mapping. We conjecture that the next to largest value of k -affinity ( f ) occurs when f is a transposition, and we are able to prove it when q = 2 , k = 2 , n ≥ 3 and when q ≥ 3 , k = 1 , n ≥ 2 .