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<p>We establish an extension of Viennot’s geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type <i>C</i> analogue of Schensted’s theorem on longest decreasing subsequences. This pairs with our results from [<CitationRef CitationID="CR3">3</CitationRef>] on Type <i>C</i> webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a 2<i>k</i>-fold tensor product of the vector representation of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak {sp}_{2n}\)</EquationSource> </InlineEquation> equals the number of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n+1)\)</EquationSource> </InlineEquation>-avoiding matchings of 2<i>k</i> points.</p>

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Decreasing subsequences and Viennot for oscillating tableaux

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月1日

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  • Elijah Bodish,
  • Ben Elias,
  • David E. V. Rose,
  • Logan Tatham

摘要

We establish an extension of Viennot’s geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type C analogue of Schensted’s theorem on longest decreasing subsequences. This pairs with our results from [3] on Type C webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a 2k-fold tensor product of the vector representation of \(\mathfrak {sp}_{2n}\) equals the number of \((n+1)\) -avoiding matchings of 2k points.