Spectra of convex hulls of matrix groups

The still-unsolved problem of determining the set of eigenvalues realized by n-by-n doubly stochastic matrices, those entrywise-nonnegative matrices with row sums and column sums equal to 1, has attracted much attention in the last century. This problem is somewhat algebraic in nature, due to a result of Birkhoff demonstrating that the set of doubly stochastic matrices is the convex hull of the permutation matrices. Here we are interested in a general matrix group G subset of GL(n)(C) and the hull spectrum HS(G) of eigenvalues realized by convex combinations of elements of G. We show that hull spectra of matrix groups share many nice properties. Moreover, we give bounds on the hull spectra of matrix groups, determine HS(G) exactly for important classes of matrix groups, and study the hull spectra of representations of abstract groups. (C) 2020 Elsevier Inc. All rights reserved.