The optimal way of estimating the spectrum of an operator is by projection onto irreducible representations of the unitary group and symmetric group. Formulated in this form by Keyl and Werner, the mathematical content of spectrum estimation has been discovered independently and in many variations by mathematicians and physicists during the last decades.

In this talk, I will show how spectrum estimation can be applied in order to study relations among operators via representation theory (and vice versa). Examples are: 1) the study of Horn's problem (addition of Hermitian operators) via Littlewood-Richardson coefficients and 2) the study of the quantum marginal problem via the Kronecker coefficients of the symmetric group.

This is a joint work with Graeme Mitchison and Aram Harrow.