Extremal Aspects of Representation Theory

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of two Hermitian matrices, given the eigenvalues of the summands.Let the summands be matrices and with eigenvaluesIn 1962, A. Horn conjectured a set of sufficient and necessary constraints for to be the spectrum of such a .These all took the form of linear inequalities in the variables , , .Some constraints are readily apparent, such asdue to the linearity of the trace, anddue to recasting the largest eigenvalue of a matrix as max{ = 1}.Other constraints are less obvious, but Horn's conjecture turned out to be true, though his list contained redundancies.Horn's inequalities all have the formwhere , , and are subsets of {1, , } of the same cardinality.If is a subset of {1, , } of cardinality , we define a partition () as follows.First enumerate the elements of in order: 1 < < .Then setConjecture 1.1.Horn conjectured that the following list of inequalities was a necessary and sufficient set of constraints for to be the spectrum of a sum = + : precisely those inequalities corresponding to , , for which (), (), and () are the eigenvalues of Hermitian matrices, the third being the sum of the first two.In other words, Horn's inequalities