<p>We classify the representation type of the descent algebras of type <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2378_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </math></EquationSource> </InlineEquation> in the positive characteristic case. The algebras have finite representation type only for a few small degrees; otherwise, they are wild. Our main reduction method relies on a surjective algebra homomorphism from a descent algebra of type <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2378_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </math></EquationSource> </InlineEquation> to another of lower degree. For small degree cases, we employ techniques from the representation theory of finite-dimensional algebras.</p>

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The representation type of the descent algebras of type \(\mathbb{A}\)

  • Karin Erdmann,
  • Kay Jin Lim

摘要

We classify the representation type of the descent algebras of type \(\mathbb{A}\) A in the positive characteristic case. The algebras have finite representation type only for a few small degrees; otherwise, they are wild. Our main reduction method relies on a surjective algebra homomorphism from a descent algebra of type \(\mathbb{A}\) A to another of lower degree. For small degree cases, we employ techniques from the representation theory of finite-dimensional algebras.