IMA

Title: Weight varieties, and projective point set

Abstract: I consider weight varieties of type A, which

are GIT quotients of the projective variety SL(n)/B,

the space of full flags in an n dimensional complex vector space,

by the action of the maximal torus T in SL(n).

A weight variety is determined by a pair (\lambda,\mu)

of weights of the group SL(n). The first weight \lambda should be dominant,

and it determines a line bundle L of SL(n)/B. The second

weight determines a lifting of the action of T on SL(n)/B to

the total space of L. If V_\lambda is the irreducible representation

of SL(n) with highest weight \lambda, let V_\lambda[\mu] denote

the \mu-th isotypic component of V_\lambda as a representation of T.

Now, the weight variety is Proj(R(\lambda,\mu)), where

R(\lambda,\mu) is the direct sum of V_{N \lambda}[N \mu] as N ranges

from 0 to infinity.

The rings R(\lambda,\mu) are not very well understood. This is in deep

contrast with the coordinate rings R(\lambda) of partial flag varieties,

which are the direct sum of V_{N \lambda}, N >= 0. The R(\lambda)

are all generated in degree one, and have just quadratic relations known

as (generalized) Plucker relations. However the rings R(\lambda,\mu) of

T invariants are not necessarily generated in degree one.

I will apply a result of my thesis and a result of Harm Derksen to show that

the rings R(\lambda,\mu) are generated in degree O(n5). Also, in the special case

that \lambda is a multiple of a fundamental weight (related to a Grassmannian),

then the bound improves to O(n2). This latter case concerns the moduli space

of n points on projective space modulo automorphisms of projective space.

This will appear in joint work with Tyrrell McAllister, where we also analyze

related toric varieties to the weight varieties. We find that certain associated toric

varieties (coming from Gelfand Tsetlin patterns)

are generated in degree *at least* O(2^n).