Representation_Theory

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title: Representation Theory permalink: wiki/Representation_Theory/ layout: wiki

Project 1: Intersection Theory (Schubert2)

Schubert2 is a package for computations in the (rational) Chow rings of smooth varieties.

Major goals for this week:

• Blow-ups: implement the ability to blow up a variety along a smooth subvariety whose intersection ring we understand
• Maps to Grassmannians: last Fall, we implemented maps to projective spaces (given by supplying a line bundle on the source). We want to extend this to maps to Grassmannians. Eventually we should be able to do maps to arbitrary flag bundles.
• Use Mike's new engine code for logg and expp, which are currently major bottlenecks for constructions of many examples.

Project 2: Representation rings of general linear groups (SchurRings)

SchurRings is a package facilitating computations in the representation rings of (products of) general linear groups.

Since the workshop in Colorado, the SchurRings package has gone through major changes: it now has the basic capabilities for computations inside representation rings of products of general linear groups: pletyhsm, SchurRings over arbitrary base rings, fast conversion between p-, h-, e-, and s-functions. This week's goal is to use the features of the package for the following applications:

• Implement a method for ``guessing" equivariant resolutions of modules with a GL-action (done).
• Compute equations for secant varieties of Segre and Veronese varieties (and other G/P's) (this is slow).

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