Reduction rules in representation theory

Abstract

A common goal in representation theory is to understand branching problems: given a representation of a group, say, we want to determine how this restricts as a representation of a subgroup of interest. In this framework, reduction rules appear as equalities in branching problems, where some quantity of interest (a dimension, a multiplicity, etc) is the same for the group and an identifiable subgroup. This identification "reduces" the required computation of these quantities to a (hopefully smaller!) computation for the subgroup.  

In the first hour, we'll review some introductory algebraic combinatorics, examining partitions, Young tableaux, and quantities encoded by them like Kostka or Littlewood-Richardson coefficients; we'll put particular focus on what kinds of data control the nonvanishing of these numbers. No background in combinatorics will be necessary. In the second hour, we will look at various reduction rules for these coefficients, and their generalizations to other representation-theoretic problems in Lie theory, as well as a common source of many such reduction rules via geometric invariant theory.