<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t_{1},\ldots ,t_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be non zero complex numbers. We consider Wada’s representation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi _{n}(t_{1},\ldots ,t_{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation><InlineEquation ID="IEq3"> <EquationSource Format="TEX">$:P_{n}\rightarrow \textrm{Gl}_{n-1}(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>:</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msub> <mtext>Gl</mtext> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when it is reducible. We then determine the irreducible representation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\hat{\varphi }_{n}(t_{1},\ldots ,t_{n}):P_{n}\rightarrow \textrm{Gl}_{n-2}(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>φ</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msub> <mtext>Gl</mtext> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and show that it is the extension of the irreducible representation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi _{n-1}(t_{1},\ldots ,t_{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Extension of Wada’s Representation of the Pure Braid Group on n Strings

期刊论文
发表日期：

2026年4月6日

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Tala S. Alaa Eddine,
Mohammad N. Abdulrahim

摘要

Let $t_{1},\ldots ,t_{n}$ $t_{1}, \dots, t_{n}$ be non zero complex numbers. We consider Wada’s representation $\varphi _{n}(t_{1},\ldots ,t_{n})$ $φ_{n} (t_{1}, \dots, t_{n})$ $:P_{n}\rightarrow \textrm{Gl}_{n-1}(\mathbb {C})$ $: P_{n} \to {Gl}_{n - 1} (C)$ when it is reducible. We then determine the irreducible representation $\hat{\varphi }_{n}(t_{1},\ldots ,t_{n}):P_{n}\rightarrow \textrm{Gl}_{n-2}(\mathbb {C})$ ${\hat{φ}}_{n} (t_{1}, \dots, t_{n}) : P_{n} \to {Gl}_{n - 2} (C)$ and show that it is the extension of the irreducible representation $\varphi _{n-1}(t_{1},\ldots ,t_{n-1})$ $φ_{n - 1} (t_{1}, \dots, t_{n - 1})$ to $P_{n}$ $P_{n}$ .