|May 03, 2022
There is a well-known bijection called the Robinson-Schensted-Knuth correspondence, which explains the Howe duality for a pair of general linear groups. It is given by a combinatorial algorithm for semistandard tableaux, which is closely related to irreducible representations of general linear groups. In this thesis, we give a combinatorial interpretation of a Howe duality associated with a pair of a symplectic group and a Lie (super)algebra. We establish a symplectic analogue of the RSK correspondence via symplectic tableaux models: spinor model and King tableaux, which are related to representations of symplectic groups and Lie (super)algebras. We introduce a symplectic analogue of jeu de taquin sliding for spinor model to define an insertion tableau in a uniform way that does not depend on the set of letters for tableaux and assign a King tableau as its recording tableau. We give new bijective proofs of well-known identities for irreducible symplectic characters as a corollary.