For a natural number n, denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> the braid group on n strings and by <InlineEquation ID="IEq2"> <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 singular braid monoid on n strings. <InlineEquation ID="IEq3"> <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> is one of the most important 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>. In [<CitationRef CitationID="CR14">14</CitationRef>], Y. Mikhalchishina classified all homogeneous 2-local representations 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> for <InlineEquation ID="IEq6"> <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 this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous 3-local representations of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. Second, we classify all homogeneous 2-local representations of <InlineEquation ID="IEq9"> <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 <InlineEquation ID="IEq10"> <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 all homogeneous 3-local representations of <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> for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.

Classification of homogeneous local representations of the singular braid monoid

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发表日期：

2025年10月29日

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Taher I. Mayassi,
Mohamad N. Nasser

摘要

For a natural number n, denote by $B_n$ $B_{n}$ the braid group on n strings and by $SM_n$ $S M_{n}$ the singular braid monoid on n strings. $SM_n$ $S M_{n}$ is one of the most important extensions of $B_n$ $B_{n}$ . In [14], Y. Mikhalchishina classified all homogeneous 2-local representations of $B_n$ $B_{n}$ for $n \ge 3$ $n \geq 3$ . In this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous 3-local representations of $B_n$ $B_{n}$ for $n \ge 4$ $n \geq 4$ . Second, we classify all homogeneous 2-local representations of $SM_n$ $S M_{n}$ for $n\ge 2$ $n \geq 2$ and all homogeneous 3-local representations of $SM_n$ $S M_{n}$ for $n\ge 4$ $n \geq 4$ .