Decreasing subsequences and Viennot for oscillating tableaux

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 [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.