Representation Theory of the Unitary and Symmetric Group with a view towards the Quantum Marginal Problem

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The quantum marginal problem asks for the compatibility of a set of partial descriptions of a quantum system. This problem is fundamental to many areas of quantum physics such as condensed matter physics and quantum information theory. Its fermionic version known as the N-representability problem is relevant to quantum chemistry where an efficient solution would allow for the calculation of properties of molecules such as their binding energies. Recently it has been shown that certain instances of the quantum marginal problem are equivalent to questions arising in the representation-theory of Lie groups and finite groups. The aim of this project is two-fold. Firstly, we wish to provide a unified treatment of the instances that have been discussed in the literature and tackle the representation-theoretic conjectures that have arisen in this context. Secondly, we wish to extend the connection between the quantum marginal problem and representation theory from specific instances to arbitrary instances, before applying our findings in quantum information theory and quantum physics more generally.