The Affinity of a Permutation of a Finite Vector Space

Details The Affinity of a Permutation of a Finite Vector Space The Affinity of a Permutation of a Finite Vector Space

2004-07-08

arxiv.org/abs/math/0407148v1

Mathematics

Combinatorics

25 pages

Scientific paper

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 establish that k-spectrum(n,q) contains 0 in the following cases: (i) q>2 and 02. The maximum of k-affinity(f) is, of course, obtained when f is any semi-affine mapping. We conjecture that the next to largest value of k-affinity(f) is when f is a transposition and we are able to prove this when q=2, k=2, n>2 and when q>2, k=1, n>1.

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