arXiv subjects

Taher I. Mayassi

Publications and source records attributed to Taher I. Mayassi.

6 recordsLinked to original sources

arXiv2026

Presentations and Representations of the Multi-Virtual Twin Group and Associated Subgroups

Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1 and n >= 2, together with two associated subgroups: the multi-virtual pure twin group M_kVPT_n and the multi-virtual semi-pure twin group M_kVHT_n.We classify all homogeneous 2-local representations of M_kVT_n into GL_n(C) for all k >= 1 and n >= 3, and show that they fall into exactly eight distinct types. We also investigate their main properties, including faithfulness and irreducibility, proving that they are generally unfaithful and providing necessary and sufficient conditions for their irreducibility.Furthermore, for certain values of k and n, we construct non-local representations of M_kVPT_n induced from those of M_kVT_n, and we determine the conditions under which these induced representations are irreducible. Finally, we present several problems for future research in this area.

math.GT↗

arXiv2025

On the classification and irreducibility of $2$-local representations of the twin group $T_n$

We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n\geqslant 2$. A complete classification is obtained, yielding three distinct families of representations. We show that each of these families is reducible by explicitly constructing one-dimensional invariant subspaces, with particular emphasis on the first family, namely $\xi_1: T_n \rightarrow \text{GL}_n(\mathbb{C})$. Passing to the corresponding quotients, we construct a reduced representation of $\xi_1$, namely $\tilde{\xi}_1: T_n \rightarrow \text{GL}_{n-1}(\mathbb{C})$. The core of the paper is that we establish, through a precise criterion, a necessary and sufficient condition for the irreducibility of the representation $\tilde{\xi}_1$.

math.RT↗

arXiv2025

Classification of Homogeneous Local Representations of the Singular Braid Monoid

For a natural number $n$, denote by $B_n$ the braid group on $n$ strings and by $SM_n$ the singular braid monoid on $n$ strings. $SM_n$ is one of the most important extensions of $B_n$. In [13], Y. Mikhalchishina classified all homogeneous $2$-local representations of $B_n$ for all $n \geq 3$. In this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous $3$-local representations of $B_n$ for all $n \geq 4$. Second, we classify all homogeneous $2$-local representations of $SM_n$ for all $n\geq 2$ and all homogeneous $3$-local representations of $SM_n$ for all $n\geq 4$.

math.RT↗

arXiv2023

Irreducible representations of the braid group $B_3$ in dimension 6

We use $q$-Pascal's triangle to define a family of representations of dimension 6 of the braid group $B_3$ on three strings. Then we give a necessary and sufficient condition for these representations to be irreducible.

math.GR↗

arXiv2021

$B_3$ block representations of dimension 6 and braid reversions

We construct a family of six dimensional block representations of the braid group $B_3$ on three strings. We show that some of these representations can be used to detect braid vertibility of some known knots and others of 9 and 10 crossings.

math.GT↗

arXiv2020

Representations of the necklace braid group {NB}}_n of dimension (n=2,3,4)

We consider the irreducible representations each of dimension 2 of the necklace braid group $\mathcal{NB}_n$ ($n=2,3,4$). We then consider the tensor product of the representations of $\mathcal{NB}_n$ ($n=2,3,4$) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $\mathcal{NB}_n$ ($n=2,3,4$) of degree 2 are unitary relative to a hermitian positive definite matrix

math.GR↗