Related papers: Non-singular word maps for linear groups

The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric…

Given an infinite linear group with a finite set of generators, we show that the shortest word length of an element of infinite order has an upper bound that depends only on the number of generators and the degree. This provides a…

We prove that a finite group is rational if and only if it has a set of permutation characters which separate conjugacy classes. It follows from this that a finite group is rational if and only if it has a representation as a permutation…

Superpermutations are words over a finite alphabet containing every permutation as a factor. Finding the minimal length of a superpermutation is still an open problem. In this article, we introduce superpermutations matrices. We establish a…

We introduce the finite-horizon first-order rank profile of a language $L \subseteq \Sigma^*$: the least quantifier rank needed by an $\mathrm{FO}[<]$ sentence to classify membership in $L$ correctly on all words of length at most $n$. The…

A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the…

Let $\Sigma_{g}$ be a closed surface of genus $g\geq 2$ and $\Gamma_{g}$ denote the fundamental group of $\Sigma_{g}$. We establish a generalization of Voiculescu's theorem on the asymptotic $*$-freeness of Haar unitary matrices from free…

We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a description…

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently…

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the…

For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank }a\leq k$. Motivated by applications to $C^{*}$-algebra theory, we investigate the…

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…

Let $\varphi:\mathbb{P}^1(\mathbb F_q)\to\mathbb{P}^1(\mathbb F_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets $\varphi^n(\mathbb{P}^1(\mathbb F_q))$ as a function of…

A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of…

A finite word $f$ is Hamming-isometric if for any two word $u$ and $v$ of same length avoiding $f$, $u$ can be transformed into $v$ by changing one by one all the letters on which $u$ differs from $v$, in such a way that all of the new…

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form…

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of…

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We…

Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal…