<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> be an affine Lie algebra with index set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{I}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">I</mi> </mrow> </math></EquationSource> </InlineEquation> = {<b>0, 1, 2,</b> <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\ldots , \varvec{n}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>…</mo> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. It is conjectured that for each Dynkin node <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{k} \in \varvec{I} \setminus \{{\textbf {0}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">k</mi> </mrow> <mo>∈</mo> <mrow> <mi mathvariant="bold-italic">I</mi> </mrow> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn mathvariant="bold">0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> the affine Lie algebra <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> has a positive geometric crystal. In this paper, we construct a positive geometric crystal for the affine Lie algebra <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq9.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{D}_{\textbf {7}}^{{\textbf {(1)}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="bold-italic">D</mi> </mrow> <mrow> <mn mathvariant="bold">7</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mn mathvariant="bold">1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> corresponding to the Dynkin spin node <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10468_2025_10325_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{k}= {\textbf {7}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">k</mi> </mrow> <mo>=</mo> <mn mathvariant="bold">7</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

$D_7^{(1)}$- Geometric Crystal at the Spin Node

期刊论文
发表日期：

2025年3月12日

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Kailash C. Misra,
Toshiki Nakashima,
Suchada Pongprasert

摘要

Let $\mathfrak {g}$ $g$ be an affine Lie algebra with index set $\varvec{I}$ $I$ = {0, 1, 2, $\ldots , \varvec{n}\}$ $\dots, n}$ . It is conjectured that for each Dynkin node $\varvec{k} \in \varvec{I} \setminus \{{\textbf {0}}\}$ $k \in I \ {0}$ the affine Lie algebra $\mathfrak {g}$ $g$ has a positive geometric crystal. In this paper, we construct a positive geometric crystal for the affine Lie algebra $\varvec{D}_{\textbf {7}}^{{\textbf {(1)}}}$ $D_{7}^{(1)}$ corresponding to the Dynkin spin node $\varvec{k}= {\textbf {7}}$ $k = 7$ .

\(D_7^{(1)}\)- Geometric Crystal at the Spin Node

摘要