We consider a 2-homogeneous bipartite distance-regular graph <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> with diameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We assume that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is not a hypercube nor a cycle. We fix a Q-polynomial ordering of the primitive idempotents of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. This Q-polynomial ordering is described using a nonzero parameter <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q \in \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> that is not a root of unity. We investigate <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> using an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-symmetric approach. In this approach one considers <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$V^{\otimes 3} = V \otimes V \otimes V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>V</mi> <mrow> <mo>⊗</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>V</mi> <mo>⊗</mo> <mi>V</mi> <mo>⊗</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> where V is the standard module of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. We construct a subspace <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$V^{\otimes 3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>V</mi> <mrow> <mo>⊗</mo> <mn>3</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> that has dimension <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>D</mi> <mo>+</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </InlineEquation>, together with six linear maps from <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>. Using these maps we turn <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> into an irreducible module for the nonstandard quantum group <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$U^\prime _q(\mathfrak {so}_6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>U</mi> <mi>q</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="fraktur">so</mi> <mn>6</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> introduced by Gavrilik and Klimyk in 1991.

2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime _q(\mathfrak {so}_6)$

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2026年2月28日

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Paul Terwilliger

摘要

We consider a 2-homogeneous bipartite distance-regular graph $\Gamma $ $Γ$ with diameter $D \ge 3$ $D \geq 3$ . We assume that $\Gamma $ $Γ$ is not a hypercube nor a cycle. We fix a Q-polynomial ordering of the primitive idempotents of $\Gamma $ $Γ$ . This Q-polynomial ordering is described using a nonzero parameter $q \in \mathbb {C}$ $q \in C$ that is not a root of unity. We investigate $\Gamma $ $Γ$ using an $S_3$ $S_{3}$ -symmetric approach. In this approach one considers $V^{\otimes 3} = V \otimes V \otimes V$ $V^{\otimes 3} = V \otimes V \otimes V$ where V is the standard module of $\Gamma $ $Γ$ . We construct a subspace $\Lambda $ $Λ$ of $V^{\otimes 3}$ $V^{\otimes 3}$ that has dimension $\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $ $(\begin{matrix} D + 3 \\ 3 \end{matrix})$ , together with six linear maps from $\Lambda $ $Λ$ to $\Lambda $ $Λ$ . Using these maps we turn $\Lambda $ $Λ$ into an irreducible module for the nonstandard quantum group $U^\prime _q(\mathfrak {so}_6)$ $U_{q}^{'} ({so}_{6})$ introduced by Gavrilik and Klimyk in 1991.

2-Homogeneous bipartite distance-regular graphs and the quantum group \(U^\prime _q(\mathfrak {so}_6)\)

摘要