University of Sydney Algebra Seminar
Sam Jeralds
Friday March 15, 12-1pm, Place: Carslaw 175
Kostant's \(V(\rho) \otimes V(\rho)\) conjecture: a tour via convex geometry
For a semisimple, complex Lie algebra \(g\), a classical question in representation theory asks how the tensor product \(V(\lambda) \otimes V(\mu)\) of two irreducible, highest weight representations \(V(\lambda)\) and \(V(\mu)\) decomposes. This is, in general, hard to predict for arbitrary \(\lambda\) and \(\mu\). For the special case of \(\lambda=\mu=\rho\), the half-sum of the positive roots of \(g\), Kostant made a conjecture which describes the irreducible components of \(V(\rho) \otimes V(\rho)\) easily and explicitly. In this talk, we'll use Kostant's conjecture as a toy example to motivate a link between branching problems in representation theory and convex geometry via families of polytopes. We also aim to describe recent work extending this conjecture to affine Kac–Moody Lie algebras (and other applications as time permits).