Determinants of sum of orbits under compact Lie group (Q765174)

Let \(G\subset U(n)\) be a compact connected subgroup of the unitary group \(U(n)\), with Lie algebra \(\mathfrak{g}\). Let \(\mathfrak{i}_{\mathfrak{u}(n)}\) be the space of \(n\times n\) Hermitian matrices and suppose \(A,B\in\mathfrak{i}_{\mathfrak{g}}\subset \mathfrak{i}_{\mathfrak{u}(n)}\). Set \[ D(A,B):=\{\det (UAU^{-1}+ VBV^{-1})\,:\, U,V\in G\}. \tag{1} \] \textit{M. Fiedler} proved [Proc. Am. Math. Soc. 30, 27--31 (1971; Zbl 0277.15010)] that for \(G=U(n)\), the set in (1) is the interval \([\min_{\sigma\in S_n}\prod_{i=1}^n(a_i+b_{\sigma(i)}),\max_{\sigma\in S_n}\prod_{i=1}^n(a_i+b_{\sigma(i)})]\), where \(a_i, b_i\) are the eigenvalues of \(A\) and \(B\), respectively. The authors determine under certain conditions, the \(m\) and \(M\) for which the set (1) is the interval \([m,M]\). In Section 3, they apply this result to the case \(G=Sp(k)\subset U(2k)\) and thus recover the extremal determinant expressions.