<p>We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to a classical question in the category <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5407_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal O(\mathfrak {gl}_r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="fraktur">gl</mi> <mi>r</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5407_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> for which the non-zero Kazhdan–Lusztig coefficients <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5407_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{\mu , \nu }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5407_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\pm 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Alternating Snake Modules and a Determinantal Formula

期刊论文
发表日期：

2025年8月1日

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Matheus Brito,
Vyjayanthi Chari

摘要

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to a classical question in the category $\mathcal O(\mathfrak {gl}_r)$ $O ({gl}_{r})$ . Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $\mu $ $μ$ for which the non-zero Kazhdan–Lusztig coefficients $c_{\mu , \nu }$ $c_{μ, ν}$ are $\pm 1$ $\pm 1$ .