Generalized matrix functions and some refinement inequalities (Q6640926)

The authors refine classical inequalities related to generalized matrix functions, specifically to permanents of doubly stochastic positive semidefinite matrices. Building on \textit{I. Schur}'s foundational work [Math. Z. 1, 184--207 (1918; JFM 46.0174.03)], the authors obtain sharper bounds and new inequalities using the Cholesky decomposition and refined the Schwarz inequalities. For example, for a subgroup \( G \subseteq S_n \), an irreducible character \( \chi \), and \( A \in M_n(\mathbb{C}) \), they prove the inequality\N\[\N\chi(e) \, d_G^\chi(AA^*) \geq |d_G^\chi(A)|^2 + \frac{\chi(e)}{t_\gamma} |d_G^\chi(A[(1, 2, \ldots, n) \mid \gamma])|^2,\N\]\Nwhere \( t_\gamma = \sum_{\sigma \in G_\gamma} \chi(\sigma) \), and \( \gamma \) is a sequence from a specific index set.\N\NAdditionally, the authors extend the results of \textit{M. Marcus} and \textit{H. Minc} [Trans. Am. Math. Soc. 116, 316--329 (1965; Zbl 0178.35902)], giving stronger lower bounds for permanent function concerning doubly stochastic positive semidefinite matrices.