A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in modulo a twisted action of the maximal torus in . An historically important example is the ring of invariants of -tuples of points in the projective plane, modulo automorphisms of the plane. We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst . In particular, we get a linear upper bound of for the degree in which is generated. We also discuss a certain toric degeneration $R'$ of that has been used to study the invariants of -tuples on the projective line. We show that this toric degeneration $R'$ ceases to be useful for higher-dimensional projective spaces. In contrast to the linear upper bound on the degrees needed to generate , $R'$ has essential generators whose degrees are exponential in . This is joint work with Benjamin J. Howard.
Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups
Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups