Ben Howard

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).