A note on the Sundaram--Stanley bijection (or, Viennot for up-down tableaux)

Abstract

We 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 sp 2n equals the number of (n + 1)-avoiding matchings of 2k points. This can be viewed as an extension of Schensted's theorem on longest decreasing subsequences. Our main tool is an extension of Viennot's geometric construction to the setting of up-down tableaux.

Triple clasp formulas for C2 webs

“…This recipe says that any matching can thus give rise to a morphism, but it is possible to rewrite such a diagram as a linear combination of diagrams associated to matchings satisfying the pattern avoidance condition [BERT21b, Theorem 5.35].7 Actually, the bijection used for type C webs, which is carefully explained in[BERT21a], is a slight variation of Sundaram's original bijection.…”