Related papers: Characterizations of Symmetrized Polydisc

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly…

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique…

We describe the centralizer of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of…

A numerical semigroup is called cyclotomic if its corresponding numerical semigroup polynomial $P_S(x)=(1-x)\sum_{s\in S}x^s$ is expressable as the product of cyclotomic polynomials. Ciolan, Garc\'ia-S\'anchez, and Moree conjectured that…

Let $P:\mathbb{C}^n\rightarrow \mathbb{C}$ be an $m$-homogeneous polynomial given by \[P(x)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}\ldots x_{j_m}.\] Defant and Schl\"uters defined a non-symmetric associated…

We characterize the group $\operatorname{Aut}(\mathbb G_2)$ for the symmetrized bidisc $\mathbb G_2:=\{(\lambda_1+\lambda_2,\lambda_1\lambda_2):|\lambda_1|,|\lambda_2|<1\}\sub set\mathbb C^2$.

Let $\Gamma\subsetneq \mathrm{Sp}_n(\mathbb{R})$ be an arithmetic subgroup of the symplectic group $\mathrm{Sp}_n(\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel…

We consider the action of the symmetric group $S_n$ on the permutahedron $\Pi_n$. We prove that if $\sigma$ is a permutation of $S_n$ which has $m$ cycles of lengths $l_1, \ldots, l_m$, then the subpolytope of $\Pi_n$ fixed by $\sigma$ has…

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson…

Here we consider the set $\Sigma_S$ of roots of power series whose coefficients lie in a given set $S$ and how such sets of roots vary as the set $S$ varies. We give an estimate of the depth that complex roots can reach into the disc, offer…

Let $M_C(G)$ be the moduli space of semistable principal $G-$bundles over a smooth curve $C$. We show that a flat degeneration of this space $M_{C_{\Gamma}}(G)$ associated to a singular stable curve $C_{\Gamma}$ contains the free group…

The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that…

Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ over the infinite field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$, and let $\circledast:\mathbb{F}G\rightarrow \mathbb{F}G$ denote the involution defined by…

Let $\Gamma_{\beta,N}$ be the $N$-part homogeneous Cantor set with $\beta\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ell\beta^{\ell-1}(1-\beta)/(N-1)$ is…

$G$ be a finite group and $A$ a $G$-graded algebra over a field $F$ of characteristic zero. We characterize the varieties of $G$-graded algebras such that the multiplicities $m_{\langle \lambda \rangle}$ appering in the $\langle n \rangle…

Quantum symmetry of a graph $C^{*}$-algebra $C^{*}(\Gamma)$ corresponding to a finite graph $\Gamma$ has been explored by several mathematicians within different categories in the past few years. In this article, we establish that there are…

If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em zigzag} in a map $M$ is a closed path which alternates choosing, at…

A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is,…

We study symmetrization procedures within the class $\mathcal S_n$ of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or $c$-convex sets via the $c$-duality $A\mapsto…

We provide a new large class of countable icc groups $\mathcal A$ for which the product rigidity result from [CdSS15] holds: if $\Gamma_1,\dots,\Gamma_n\in\mathcal A$ and $\Lambda$ is any group such that…