Let t be indeterminate and let <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\rho : B_n \rightarrow GL_{m}(\mathbb {Z}[t^{\pm 1}])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>:</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mi>G</mi> <msub> <mi>L</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be a k-local representation of the braid group <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k=m-n+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We consider two types of extensions of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> to the singular braid monoid <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. The first type is called the k-local extension, which is a new concept we defined in this paper. The second type is called the <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-type extension, which is given by Bardakov, Chbili, and Kozlovskaya in (Mediterr. J. Math. 21: 180, 2024). In order to obtain a relation between these two types of extensions, we consider two homogeneous 2-local representations of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, namely <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\rho _B: B_n \rightarrow GL_{n}(\mathbb {Z}[t^{\pm 1}])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mi>B</mi> </msub> <mo>:</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mi>G</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\rho _S: B_n \rightarrow GL_{n}(\mathbb {Z}[t^{\pm 1}])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mi>S</mi> </msub> <mo>:</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mi>G</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and a homogeneous 3-local representation of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, namely <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\rho _F: B_n \rightarrow GL_{n+1}(\mathbb {Z}[t^{\pm 1}])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mi>F</mi> </msub> <mo>:</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We study the relation between the two types of extensions to <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of these three representations of <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. In fact, we prove that every homogeneous 2-local extension of <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\rho _B$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation> is also a <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-type extension for all <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>; which is not the case for <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$\rho _S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation>. Also, we prove that every homogeneous 3-local extension of <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\rho _F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-type extension for all <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, we study, in the case <InlineEquation ID="IEq28"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, the faithfulness of the complex specialization of all 2-local extensions of <InlineEquation ID="IEq29"> <EquationSource Format="TEX">$\rho _B$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq30"> <EquationSource Format="TEX">$\rho _S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation>.

Local Extensions and $\Phi $-Type Extensions of Some Local Representations of the Braid Group $B_n$ to the Singular Braid Monoid $SM_n$

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发表日期：

2025年4月16日

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Mohamad N. Nasser

摘要

Let t be indeterminate and let $\rho : B_n \rightarrow GL_{m}(\mathbb {Z}[t^{\pm 1}])$ $ρ : B_{n} \to G L_{m} (Z [t^{\pm 1}])$ be a k-local representation of the braid group $B_n$ $B_{n}$ , where $k=m-n+2$ $k = m - n + 2$ . We consider two types of extensions of $\rho $ $ρ$ to the singular braid monoid $SM_n$ $S M_{n}$ . The first type is called the k-local extension, which is a new concept we defined in this paper. The second type is called the $\Phi $ $Φ$ -type extension, which is given by Bardakov, Chbili, and Kozlovskaya in (Mediterr. J. Math. 21: 180, 2024). In order to obtain a relation between these two types of extensions, we consider two homogeneous 2-local representations of $B_n$ $B_{n}$ , namely $\rho _B: B_n \rightarrow GL_{n}(\mathbb {Z}[t^{\pm 1}])$ $ρ_{B} : B_{n} \to G L_{n} (Z [t^{\pm 1}])$ and $\rho _S: B_n \rightarrow GL_{n}(\mathbb {Z}[t^{\pm 1}])$ $ρ_{S} : B_{n} \to G L_{n} (Z [t^{\pm 1}])$ , and a homogeneous 3-local representation of $B_n$ $B_{n}$ , namely $\rho _F: B_n \rightarrow GL_{n+1}(\mathbb {Z}[t^{\pm 1}])$ $ρ_{F} : B_{n} \to G L_{n + 1} (Z [t^{\pm 1}])$ . We study the relation between the two types of extensions to $SM_n$ $S M_{n}$ of these three representations of $B_n$ $B_{n}$ . In fact, we prove that every homogeneous 2-local extension of $\rho _B$ $ρ_{B}$ is also a $\Phi $ $Φ$ -type extension for all $n\ge 2$ $n \geq 2$ ; which is not the case for $\rho _S$ $ρ_{S}$ . Also, we prove that every homogeneous 3-local extension of $\rho _F$ $ρ_{F}$ is a $\Phi $ $Φ$ -type extension for all $n\ge 3$ $n \geq 3$ . In addition, we study, in the case $n=2$ $n = 2$ , the faithfulness of the complex specialization of all 2-local extensions of $\rho _B$ $ρ_{B}$ and $\rho _S$ $ρ_{S}$ .

Local Extensions and \(\Phi \)-Type Extensions of Some Local Representations of the Braid Group \(B_n\) to the Singular Braid Monoid \(SM_n\)

摘要