# Orbit closures in nullcones of group representations

### Summary

Describing the geometry and singularities of orbit closures in the nullcones of representations of reductive groups, generalizing the known results about adjoint representations on Lie algebras and other parts of classical invariant theory.

### Supervisor(s)

Associate Professor Anthony Henderson

### Research Location

School of Mathematics and Statistics

### Program Type

PHD

### Synopsis

In any representation of a reductive algebraic group (say, over the complex numbers), the Hilbert nullcone consists of the vectors whose orbit closure contains zero. For some representations (not necessarily irreducible), the group acts with finitely many orbits on the nullcone, and the singularities of the orbit closures have connections to other parts of algebra. The classic example is the nilpotent cone in the Lie algebra of the group, which is related to representations of Weyl groups and Hecke algebras vis the Springer correspondence. Recently-studied examples are the exotic and enhanced nilpotent cones. There are many basic questions which remain unanswered even for quite small representations (notably of exceptional groups): for example, orbit classification, closure partial order, resolutions of singularities, intersection cohomology.

### Additional Information

A good Honours degree (or equivalent) majoring in some algebraic area of pure mathematics is essential. Prior knowledge of group representation theory and algebraic geometry is highly desirable.

The School of Mathematics and Statistics has a large and active Algebra research group including (as of 2009) seven continuing academic and research staff members, three postdoctoral researchers and ten PhD students. The weekly Algebra seminar showcases the research of the group and its many visitors.

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### Keywords

Algebra; representation theory; algebraic groups; orbits; nullcones; geometry

### Opportunity ID

The opportunity ID for this research opportunity is: **1045**

### Other opportunities with Associate Professor Anthony Henderson