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Mohamad N. Nasser

Publications and source records attributed to Mohamad N. Nasser.

At least 19 recordsLinked to original sources
arXiv2026

On the irreducibility of singular braid group representations

In this article, we investigate the irreducibility of representations of the singular braid group on $n$ strands, denoted by $SB_n$. We first show that all representations $ρ: SB_2 \longrightarrow \mathrm{GL}_2(\mathbb{C})$ are reducible. We then determine all irreducible representations $ρ: SB_3 \longrightarrow \mathrm{GL}_2(\mathbb{C})$ and construct several families of irreducible representations $ρ: SB_3 \longrightarrow \mathrm{GL}_3(\mathbb{C})$. Finally, we study the irreducibility of homogeneous local representations of $SB_n$ for all $n \geq 3$.

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arXiv2026

On representations of the triplet group and some of its extensions

In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, together with their extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation $Θ: L_n \longrightarrow\mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \longrightarrow \mathrm{GL}_{n}(\mathbb{Z}[t^{\pm 1}])$, where $t$ is an indeterminate. For the representation $μ$, we completely study its faithfulness and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all complex non-homogeneous $2$-local representations of $L_3$, establishing connections with the complex specialization of the representation $μ$. Finally, we examine extensions of $L_n$ representations to $VL_n$ and $WL_n$, proving their existence, classifying non-trivial complex homogeneous $2$-local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question concerning further extensions of representations of $L_n$ to $VL_n$ and $WL_n$.

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arXiv2026

On the structure of the singular triplet monoid and its virtual extension

In this article, we introduce two new algebraic structures associated with the triplet group on $n$ strands, $L_n$: the singular triplet monoid $SLM_n$ and its virtual extension $VSLM_n$, defined in analogy with the singular braid monoid and the virtual singular braid monoid. We begin by presenting these monoids in terms of generators and relations, and then derive several alternative presentations of $VSLM_n$. Second, we investigate the problem of extending representations of $L_n$ to these monoids. Two extension methods are developed: the $k$-local type extension, which applies to $k$-local representations, and the $Φ$-type extension, which applies to representations satisfying suitable commutativity conditions. We show that every $2$-local representation of $L_n$ admits extensions to both $SLM_n$ and $VSLM_n$ via the two methods. As an application, we consider a specific representation $μ: L_n \longrightarrow \mathrm{GL}_n(\mathbb{Z}[t^{\pm1}])$ introduced recently by Nasser et al. We explicitly determine all homogeneous $2$-local extensions of $μ$ to $SLM_n$ and $VSLM_n$, and compute the corresponding $Φ$-type extensions. Furthermore, we compare these two extension methods, showing that they coincide for $SLM_n$ under suitable parameter conditions, while they do not coincide for $VSLM_n$. These results provide a systematic framework for extending representations of $L_n$ to $SLM_n$ and $VSLM_n$.

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arXiv2026

Matrix representations of the twisted virtual braid group and its extensions

This paper classifies complex local representations of the twisted virtual braid group, $TVB_2$, into $\mathrm{GL}_3(\mathbb{C})$. It shows that such representations fall into eight types, all of which are unfaithful and reducible to a two-dimensional representation. Further reducibility to a one-dimensional representation is analyzed for specific types. The paper also examines complex homogeneous local representations of $TVB_n$ into $\mathrm{GL}_{n+1}(\mathbb{C})$ for $n \geq 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, $STVB_2$, into $\mathrm{M}_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of $TVB_2$ representations to $STVB_2$ conform to a $Φ$-type extension.

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arXiv2026

Local Representations of the Flat Virtual Braid Group

We prove that any complex local representation of the flat virtual braid group, $FVB_2$, into $GL_2(\mathbb{C})$, has one of the types $λ_i: FVB_2 \rightarrow GL_2(\mathbb{C})$, $1\leq i\leq 12$. We find necessary and sufficient conditions that guarantee the irreducibility of representations of type $λ_i$, $1\leq i\leq 5$, and we prove that representations of type $λ_i$, $6\leq i\leq 12$, are reducible. Regarding faithfulness, we find necessary and sufficient conditions for representations of type $λ_6$ or $λ_7$ to be faithful. Moreover, we give sufficient conditions for representations of type $λ_1$, $λ_2$, or $λ_4$ to be unfaithful, and we show that representations of type $λ_i$, $i=3, 5, 8, 9, 10, 11, 12$ are unfaithful. We prove that any complex homogeneous local representations of the flat virtual braid group, $FVB_n$, into $GL_{n}(\mathbb{C})$, for $n\geq 3$, has one of the types $γ_i: FVB_n \rightarrow GL_n(\mathbb{C})$, $i=1, 2$. We then prove that representations of type $γ_1: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 6$, while representations of type $γ_2: FVB_n \rightarrow GL_n(\mathbb{C})$ are irreducible if and only if $b\neq y$, for $n\geq 3$. Then, we show that representations of type $γ_1$ are unfaithful for $n\geq 3$ and that representations of type $γ_2$ are unfaithful if $y=b$. Furthermore, we prove that any complex homogeneous local representation of the flat virtual braid group, $FVB_n$, into $GL_{n+1}(\mathbb{C})$, for all $n\geq 4$, has one of the types $δ_i: FVB_n \rightarrow GL_{n+1}(\mathbb{C})$, $1\leq i\leq 8$. We prove that these representations are reducible for $n\geq 10$. Then, we show that representations of types $δ_i$, $i\neq 5, 6$, are unfaithful, while representations of types $δ_5$ or $δ_6$ are unfaithful if $x=y$.

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arXiv2026

Multi-welded twin groups

For $k\geq 1$ and $n\geq 2$, we introduce the multi-welded twin group $M_kWT_n$, a natural welded analogue of the multi-virtual twin group. We show that $M_kWT_n$ arises naturally as a quotient of the universal welded braid group $UW_n(k)$, placing it within the unified framework of universal virtual and welded braid-type groups. We establish natural quotient maps relating $M_kWT_n$ to the multi-virtual twin group $M_kVT_n$, the welded twin group $WT_n$, and the corresponding virtual and welded braid-type groups. Several structural properties of $M_kWT_n$ are obtained. In particular, we compute its abelianization, prove that its commutator subgroup is perfect for $n\ge5$, and show that the symmetric group $S_n$ is its smallest non-abelian finite quotient. We also investigate the representation theory of $M_kWT_n$. In fact, we classify all non-trivial complex homogeneous $2$-local representations of $M_kWT_n$, showing that only one family survives under the additional twin and welded relations. Furthermore, we classify all non-trivial complex homogeneous $3$-local representations of $M_2WT_n$. We further investigate the reducibility and faithfulness properties of both the $2$-local and $3$-local representations.

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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.

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arXiv2026

On Universal Virtual and Welded Braid Groups and Their Linear Representations

We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex homogeneous $2$-local representations for all $n\geq 3$ and $c\geq 1$ (unique up to equivalence) and complex homogeneous $3$-local representations for all $n\geq 4$ and $c=2$ (four distinct families). We then introduce the universal welded braid group $UW_n(c)$ as a quotient of $UV_n(c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $UW_n(c)$ has abelianization $\mathbb{Z}^c \oplus \mathbb{Z}_2$, perfect commutator subgroup for $n \geq 5$, trivial center, and $S_n$ as its smallest non-abelian finite quotient. Finally, we classify and study the complex homogeneous $2$-local representations of $UW_n(c)$ for all $n\geq 3$ and $c\geq 1$, obtaining three distinct families.

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arXiv2026

Twin groups representations

We construct two representations of the twin group $T_n, n\geq 2$, namely $η_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $η_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $η_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $η_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions.

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arXiv2026

Insights on the homogeneous $3$-local representations of the twin groups

We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $VT_n$, and the welded twin group $WT_n$, for all $n\geq 4$. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of $T_n$ to be irreducible in the case $n=4$. The obtained results provide insights into the algebraic structure of these three groups.

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arXiv2026

Algebraic and topological aspects of the singular twin group and its representations

In this article, we introduce the singular twin monoid and its corresponding group, constructed from both algebraic and topological perspectives. We then classify all complex homogeneous $2$-local representations of this constructed group. Moreover, we study the irreducibility of these representations and provide clear conditions under which irreducibility holds. Our results give a structured approach to understanding this new algebraic object and its representations.

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arXiv2026

The virtual singular twin monoid and group: presentations and representations

In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures extend the twin group $T_n$ in close analogy to how the virtual singular braid monoid and virtual singular braid group extend the classical braid group. We then construct and study representations of the group $VST_n$, for $n \geq 3$, focusing in particular on extending the representations $η_1$ and $η_2$ of $T_n$, introduced by M. Nasser, to $VST_n$ via the $2$-local extension method. To analyze the resulting representations, $η_1'$ and $η_2'$, and their properties, we establish necessary and sufficient conditions for irreducibility and show that both $η_1'$ and $η_2'$ are unfaithful. Additionally, we classify all complex homogeneous $2$-local representations of $VST_n$ for every integer $n\geq 3$, providing a foundation for further investigation into representations of $VST_n$.

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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 $ξ_1: T_n \rightarrow \text{GL}_n(\mathbb{C})$. Passing to the corresponding quotients, we construct a reduced representation of $ξ_1$, namely $\tildeξ_1: T_n \rightarrow \text{GL}_{n-1}(\mathbb{C})$. The core of the paper is that we establish, through a precise criterion, a necessary and sufficient condition for the irreducibility of the representation $\tildeξ_1$.

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arXiv2025

On representations of the multi-virtual braid group $M_kVB_n$ and the multi-welded braid group $M_kWB_n$

This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_kVB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, showing that such representations fall into exactly $2^{k+1}+1$ distinct types, out of which except three all are unfaithful. In addition, this paper investigates complex homogeneous $2$-local representations of the multiple welded braid group $M_kWB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, identifying $3 \cdot 2^{k-1} +1$ representations. Moreover, the article includes a construction of a non-local representation of $M_2WB_3$ that extends the known LKB representation of the braid group on $3$ strands, namely $B_3$, making a path towards constructing non-local representations of $M_kWB_n$ in general.

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arXiv2025

On extensions of the standard representation of the braid group to the singular braid group

For an integer $n \geq 2$, set $B_n$ to be the braid group on $n$ strands and $SB_n$ to be the singular braid group on $n$ strands. $SB_n$ is one of the important group extensions of $B_n$ that appeared in 1998. Our aim in this paper is to extend the well-known standard representation of $B_n$, namely $ρ_S:B_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, to $SB_n$, for all $n \geq 2$, and to investigate the characteristics of these extended representations as well. The first major finding in our paper is that we determine the form of all representations of $SB_n$, namely $ρ'_S: SB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$, for all $n\geq 2$. The second major finding is that we find necessary and sufficient conditions for the irreduciblity of the representations of the form $ρ'_S$ of $SB_n$, for all $n\geq 2$. We prove that, for $t\neq 1$, the representations of the form $ρ'_S$ are irreducible and, for $t=1$, the representations of the form $ρ'_S$ are irreducible if and only if $a+c\neq 1.$ The third major result is that we consider the virtual singular braid group on $n$ strands, $VSB_n$, which is a group extension of both $B_n$ and $SB_n$, and we determine the form of all representations $ρ''_S: VSB_2 \to GL_2(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$; making a path toward finding the form of all representations $ρ''_S: VSB_n \to GL_n(\mathbb{Z}[t^{\pm 1}])$, that extend $ρ_S$ and $ρ'_S$, for all $n\geq 3$.

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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 $n\geq 4$.

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arXiv2024

On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$

For $n\geq 2$, let $G_n$ be a group and let $ρ: B_n\rightarrow G_n$ be a representation of the braid group $B_n$. For a field $\mathbb{K}$ and $a,b,c\in \mathbb{K}$, Bardakov, Chbili, and Kozlovskaya extend the representation $ρ$ to a family of representations $Φ_{a,b,c}:SM_n \rightarrow \mathbb{K}[G_n]$ of the singular braid monoid $SM_n$, where $\mathbb{K}[G_n]$ is the group algebra of $G_n$ over $\mathbb{K}$. In this paper, we study the faithfulness of the family of representations $Φ_{a,b,c}$ in some cases. First, we find necessary and sufficient conditions of the families $Φ_{a,0,0}, Φ_{0,b,0}$ and $Φ_{0,0,c}$ for all $n\geq 2$ to be unfaithful, where $a,b,c \in \mathbb{K}^*$. Second, we consider the case $n=2$ and we find the nature of $\ker(Φ_{a,b,c})$ if $Φ_{a,b,c}$ is unfaithful. Moreover, we show that there exist some families $Φ_{a,b,c}$ that have trivial kernel in the case $n=2$. Also, we find the shape of the possible elements in $\ker(Φ_{a,b,c})$ for all $n\geq 3$ when the kernel of ${Φ_{a,b,c}|}_{SM_2}$ is nontrivial.

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arXiv2023

The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$

Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $ρ$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation $ρ$ is unfaithful for $n \geq 5$. The cases $n=3,4$ remain open. M. N. Nasser and M. N. Abdulrahim made attempts towards the faithfulness of $ρ$ in the case $n=3$. In this work, we prove that $ρ$ is unfaithful in the both cases $n=3$ and $n=4$.

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