algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety flag variety; Grassmannian variety; Schubert variety; symmetric group; algebraic group; Lie algebra; Schur-Weyl duality; singular locus; Chevalley group; standard monomial theory; general linear group; special linear group; Borel subgroup; parabolic subgroup; determinantal variety; commutative algebra; algebraic geometry; character; semisimple ring; finite group; toric variety; invariant theory; tableaux; irreducible module; Bruhat decomposition; sheaf cohomology 