Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups

TL;DR

This paper establishes degree bounds for generators of type-A weight rings, showing they are generated below the Krull dimension, but also reveals that Gelfand--Tsetlin semigroups can have exponentially large generators.

Contribution

It proves that weight rings in type A are generated by elements of degree less than the Krull dimension, providing a new bound, and analyzes the complexity of Gelfand--Tsetlin semigroups.

Findings

Weight rings are generated by elements of degree less than the Krull dimension.

The Krull dimension of these rings is at most quadratic in n.

Gelfand--Tsetlin semigroups can have essential generators of exponential degree.

Abstract

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\C^n$ modulo a twisted action of the maximal torus in $\SL(n,\C)$. 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 $O(n^2)$. On the other hand, we show that the associated semigroup of Gelfand--Tsetlin patterns can have an essential generator of degree exponential in $n$.