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This paper aims to focus on Richardson varieties on symplectic groups,especially their combinatorial characterization and defining equations.Schubert varieties and opposite Schubert varieties have profound significancein the study of generalized flag varieties which are not only research objectsin algebraic geometry but also ones in representation theory. A more generalresearch object is Richardson variety, which is obtained by the intersection ofa Schubert variety and an opposite Schubert variety. The structure ofRichardson variety on Grassmannian and its combinatorial characterization arewell known, and there are also similar method on quotients of symplecticgroups. In the first part of this paper, we calculate the orbit of thesymplectic group action, and then rigorously give a method to describe thecorresponding quotient by using the nesting subspace sequence of the linearspace, i.e. flags. At the same time, the flag is used to describe the Schubertvariety and Richardson variety on quotient of symplectic group. The flagvarieties of Sp_{2n}(k)/P_d can be viewed as closed subvarieties ofGrassmannian. Using the standard monomial theory, we obtain the generators ofits ideal, i.e. its defining equations, in homogeneous coordinate ring ofGrassmannian. Furthermore, we prove several properties of the type C standardmonomial on the symplectic group flag variety. Defining equations of Richardsonvarieties on Sp_{2n}(k)/P_d are given as well.

The defining equations of a class of Richardson and flag varieties on Sp$_{2n}(k)$