Decreasing subsequences and Viennot for oscillating tableaux

TL;DR

This paper extends Viennot's shadow line construction to oscillating tableaux, providing a new proof of the Type C analogue of Schensted's theorem and linking it to invariant vector spaces in symplectic tensor products.

Contribution

It introduces an extension of Viennot's geometric construction to oscillating tableaux and offers a new proof of a key theorem in Type C combinatorics.

Findings

Extended Viennot's construction to oscillating tableaux

Provided a new proof of Type C Schensted's theorem

Connected combinatorial structures to invariant vector spaces in symplectic representations

Abstract

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 arXiv:2103.14997v1 [math.RT] 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.