For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.
The group of polynomial automorphisms of the affine n-space is an interesting large group. A slightly simpler group is its subgroup of tame automorphisms. Natural problems about these groups include the existence of normal subgroups, the classification of finite subgroups, the Tits alternative, and the possible dynamic...
For a (quasi)projective variety $X$ and an automorphism $f$ of $X$, we define moduli spaces of twisted endomorphisms of sheaves on $X$. If the automorphism $f$ has finite order, we construct a sheaf of noncommutative algebras $\mathcal{A}_0(f)$ and show that sheaves with a twisted endomorphism can be described as $\mat...
We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie a...
Michael Chitayat, D. Daigle, Andriy Regeta· 0 citations
We classify abstract automorphisms of the elementary and full twisted Chevalley groups of type ${}^2E_6$ over commutative local rings with involution and $2$ invertible, for both isogeny types. Every automorphism is standard, with no invertibility assumption on $3$. In the adjoint case, it has a unique expression as a...
We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms cons...
Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field with $q$ elements of characteristic $p$ large enough. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. In the paper, we provide a classification of such orbits of pre-maximal dime...
M. Ignatev, Mikhail Venchakov· 1 citation
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