Research shows a new proof of the Type C analogue of Schensted’s theorem on decreasing subsequences, implying important results about invariant vectors.
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 sp₂ₙ equals the number of $$(n+1)$$ -avoiding matchings of 2k points.
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Bodish et al. (2026) studied this question.
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