Symbolic Powers of Invariant Ideals

The symbolic powers first appear in Krul's principal ideal theorem.  Though defined algebraically, symbolic powers connect geometry and algebra. It reveals many geometric properties as well. One of the properties is finding minimal degree hypersurfaces vanishing at a given variety with a certain multiplicity. Even though they are extremely useful, it is difficult to find a tangible description of the symbolic powers. They are known only in a few cases. In this talk, I will describe symbolic powers and different open problems related to them. I will also describe some numerical invariants connected with symbolic powers. I will give a tangible description of the symbolic powers of invariant ideals arising via young diagrams. I will discuss the results from my joint with Sudipta Das and Alexandra Seceleanu.