Schubert calculus, adjoint representation and moment polytopes

Abstract

Let Hν denote the irreducible representation of the special unitary group SU(n) corresponding to Young diagram ν and let g = su(n) denote the Lie algebra of SU(n). One can show that Hν appears in the symmetric algebra S∗ g if and only if n divides the size of the Young diagram ν. Kostant’s problem asks what is the least number N such that Hν appear in SN g. The moment polytope of the adjoint representation is the polytope generated by the normalized weights ν˜ such that Hν appears in S∗ g and it helps to put lower bounds on number N in the Kostant’s problem. In this thesis, we compute the moment polytope of the adjoint representation of SU(n) for n ≤ 9 using the solutions of the classical spectral problem and so-called ν-representability problem.