Related papers: On the Automorphism Group of a Binary Form Associa…

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. We estimate the area $A_F$ bounded by the curve $|F(x, y)| = 1$ for four families of binary forms. The first two families that we are interested in are…

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true…

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the…

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps…

Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar\'e series of $G$, meaning the series $f(z)=1+c_1z+c_2z^2+...$ whose coefficients are multiplicities of 1 into tensor powers…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is…

It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.

We describe a set of defining relations of the tame automorphism group TA_3(F) of the polynomial algebra F[x_1,x_2,x_3] in variables x_1,x_2,x_3 over an arbitrary field F of characteristic 0.

We consider the Macdonald group $\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}\rangle$ and its Sylow 2-subgroup $J=\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell},…

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional…

We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for…

In this paper, we introduce notions of partitionability and characteristic sets of homogeneous polynomials and give a complete classification of groups faithfully acting on smooth cubic fivefolds. Specifically, we prove that there exist 20…

Let $K[x,y]$ be the polynomial algebra in two variables over an algebraically closed field $K$. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms $(f,g)$…

The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={\rm Der\,}A_n$ are studied in this paper. It is well-known that ${\rm Aut\,}A_n$ for…

Given a binary form $F \in \mathbb{Z}[X, Y]$, we define its value set to be $\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. Let $F, G \in \mathbb{Z}[X, Y]$ be two binary forms of degree $d \geq 3$ and with non-zero discriminant. In a series of…

Automorphism groups of $2$-groups of coclass at most $3$ are investigated.

We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure…

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$ with automorphism groups isomorphic to the dihedral group of cardinality $6$ or $12$. We characterize exactly when $F$ and $G$ have the same value set, i.e. $F(\mathbb{Z}^2)…

For any choice of a basis $\cal A$ the free group $F_N$ of finite rank $N \geq 2$ can be canonically identified with the set $F(\cal A)$ of reduced words in $\cal A\cup \cal A^{-1}$. However, such a word $w \in F(\cal A)$ admits a second…