Richardson Varieties On Symplectic Groups And Their Geometric Properties

This paper aims to focus on Richardson varieties on symplectic groups and their geometric properties.Schubert varieties and opposite Schubert varieties have profound significance in the study of generalized flag varieties.Generalized flag varieties are not only research objects in algebraic geometry,they are also important research objects in representation theory.A more general research object is Richardson variety,which is obtained by the intersection of a Schubert variety and an opposite Schubert variety.Firstly,for the cases of the general linear groups and symplectic groups,this paper calculates the orbit of the linear algebraic group act,and then clearly gives a method to describe the corresponding quotient by using the nesting subspace sequence of the linear space(i.e.flag).At the same time,the flag is used to describe the Schubert variety and Richardson variety on the general linear group and symplectic group.The flag varieties of Sp2n(k)/Pd can be viewed as closed subvarieties of Grassmannian.Using the standard monomial theory,we obtain the generators of its ideal,i.e.its defining equations,in homogeneous coordinate ring of Grassmannian.Furthermore,we prove several properties of the C type standard monomial on the symplectic group flag.Defining equations of Richardson varieties on Sp2n(k)/Pd are given as well.Finally,this paper reviews the work of Victor Kreiman,V.Lakshmibai and Sara Billey on the singular loci of the Schubert and Richardson varieties on Sp2n(k)/Pn,supplemented with some details and proof.