2009

Recent Trends in Invariant Theory

by

Prof. Roger Howe
Mathematics Department, Yale University

on

February 10, 2009

at

4PM, L H ? 1, Department of Mathematics
Indian Institute of Science, Bangalore

Abstract

Since the early days of invariant theory, an important goal has been to describe the ring of all invariant polynomial functions for a given group action on a vector space. However, progress has been limited by the fact that aside from a restricted number of favorable examples, these rings tend to have rather complicated structure. In recent years, the value of using the idea of toric deformation has emerged as a promising tool in invariant theory. Toric deformation allows one to replace a complicated ring by a simpler one that still carries most or all of the numerical and combinatorial information that one wants from the ring of invariants. The simpler rings can be described in terms of {\it lattice cones}:the collection of integral points in a convex polyhedral cone in Euclidean space. This gives rise to a theory with a geometric flavor in which numbers of interest, such as dimensions of eigenspaces, are described the the collection of integral points in a convex polyhedron. Toric deformations promise to provide a systematic understanding of topics that have been the submect of intense and continuing study since the early 20th century. The goal of this talk is to provide an overview of this new approach to invariant theory.