We consider in this study the known Wada representations of the braid group on n strings, denoted as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. A complete study on three important types of the Wada representations has been done by Abdulrahim and Tahan. On the other hand, Mikhalchishina extended the Wada representations of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to the virtual and the welded braid groups, which are famous extensions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. As extensions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, which are monoids, we consider in this paper the singular braid monoid on n strings, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and the virtual singular braid monoid on n strings, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$VSM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We aim to look for extensions of the Wada representations of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to these two monoids for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and to recognize some characteristics of these extensions as well. Specifically, as the Wada representation of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of type 1 is homogeneous 2-local, we consider its 2-local extensions to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$VSM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Another family of extensions of the Wada representation of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of type 1, to <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, to be considered in our work, are the <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-type extensions, for all <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Our first result is the classification of all forms of these two types of extensions. The second result is the study of the irreducibility of these representations on <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and the third result is the study of their faithfulness in the case <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

A note on extending the Wada representations of the braid group to the singular braid monoid

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2026年3月17日

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Mohamad N. Nasser

摘要

We consider in this study the known Wada representations of the braid group on n strings, denoted as $B_n$ $B_{n}$ , for $n\ge 2$ $n \geq 2$ . A complete study on three important types of the Wada representations has been done by Abdulrahim and Tahan. On the other hand, Mikhalchishina extended the Wada representations of $B_n$ $B_{n}$ to the virtual and the welded braid groups, which are famous extensions of $B_n$ $B_{n}$ . As extensions of $B_n$ $B_{n}$ , which are monoids, we consider in this paper the singular braid monoid on n strings, $SM_n$ $S M_{n}$ , and the virtual singular braid monoid on n strings, $VSM_n$ $V S M_{n}$ . We aim to look for extensions of the Wada representations of $B_n$ $B_{n}$ to these two monoids for all $n\ge 2$ $n \geq 2$ , and to recognize some characteristics of these extensions as well. Specifically, as the Wada representation of $B_n$ $B_{n}$ of type 1 is homogeneous 2-local, we consider its 2-local extensions to $SM_n$ $S M_{n}$ and to $VSM_n$ $V S M_{n}$ for all $n\ge 2$ $n \geq 2$ . Another family of extensions of the Wada representation of $B_n$ $B_{n}$ of type 1, to $SM_n$ $S M_{n}$ , to be considered in our work, are the $\Phi $ $Φ$ -type extensions, for all $n\ge 2$ $n \geq 2$ . Our first result is the classification of all forms of these two types of extensions. The second result is the study of the irreducibility of these representations on $SM_n$ $S M_{n}$ for all $n \ge 2$ $n \geq 2$ , and the third result is the study of their faithfulness in the case $n=2$ $n = 2$ .