Monday, February 27, 2006
2:45 pm, Small Auditorium (Science Building, 1st floor)

David Vogan

MIT

Branching Laws For Group Representations

Abstract: Representation theory concerns group actions on vector spaces, so some of its most fundamental questions concern the dimensions of those vector spaces. If G is a group and K is a subgroup, a branching law from G to K describes how often each irreducible representation of K appears in an irreducible representation of G. The answer is a "branching matrix" of non-negative integers. I will explain that in many interesting cases, this (infinite) matrix may be regarded as square, upper triangular, and having 1's on the diagonal. It therefore makes sense (both theoretically and computationally) to invert the branching matrix. One can sometimes write a very simple closed formula for the inverse matrix. Computing the branching law then becomes the linear algebra problem of inverting an upper triangular matrix.