On Permutations Induced by Tame Automorphisms Over Finite Fields

Abstract

Abstract. The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism.

Generalization of a counterexample to Derksen’s theorem in characteristic two

“…, X n )) ∈ π 2 (Aff n (F 2 )), we have π 2 (Aff n (F 2 )) = π 2 ( Aff n (F 2 ), ) = π 2 (DA n (F 2 )). Then by (Hakuta 2018, Corollary 2), the containment π 2 (Aff n (F 2 )) = π 2 (DA n (F 2 )) ⊂ Alt(F n 2 ) holds for each n ≥ 3. On the other hand, from (Maubach 2001, Theorem 2.3(ii)), we have π 2 (TA n (F 2 )) = Sym(F n 2 ).…”

Sign of Permutation Induced by Nagata Automorphism over Finite Fields

“…Here, we give a more simple proof of Lemma 2 and Lemma 3. It follows immediately from (Hakuta, 2017a, Main Theorem 1) that sgn(π q (λ)) = 1. Again from (Hakuta, 2017a, Main Theorem 1), we have sgn(π q (λ)) =        1 (q is odd or q = 2 m and m ≥ 2) , −1 (q = 2) .…”

“…This question is partially solved in the case of the Anick automorphism and the Nagata-Anick automorphism by Hakuta (Hakuta, 2017b, Main Theorem 1 and Main Theorem 2). Hakuta also derives the sign of the permutations induced by the affine automorphisms and the elementary automorphisms (Hakuta, 2017a, Main Theorem 1 and Main Theorem 2). Moreover, for a given tame automorphism over F q , we can compute the sign of the permutation induced by the tame automorphism over F q under the knowledge of a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms (Hakuta, 2017a, Corollary 5).…”