CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups Tyrrell McAllister, University of Wyoming
February 14, 2012 Math 350 2-3pm	Abstract A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\mathbb{C}^n$ modulo a twisted action of the maximal torus in $SL(n,\mathbb{C})$. An historically important example is the ring $R$ of invariants of $n$-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 $O(n^2)$. In particular, we get a linear upper bound of $2n ? 8$ for the degree in which $R$ is generated. We also discuss a certain toric degeneration $R'$ of $R$ that has been used to study the invariants of $n$-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 $R$, $R'$ has essential generators whose degrees are exponential in $n$. This is joint work with Benjamin J. Howard.

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

Tyrrell McAllister, University of Wyoming

February 14, 2012

Math 350

2-3pm

Abstract

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\mathbb{C}^n$ modulo a twisted action of the maximal torus in $SL(n,\mathbb{C})$. An historically important example is the ring $R$ of invariants of $n$-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 $O(n^2)$. In particular, we get a linear upper bound of $2n ? 8$ for the degree in which $R$ is generated. We also discuss a certain toric degeneration $R'$ of $R$ that has been used to study the invariants of $n$-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 $R$, $R'$ has essential generators whose degrees are exponential in $n$. This is joint work with Benjamin J. Howard.