Some remarks on permanental dominance conjecture (Q6614414)

For a finite group \(G\) embedded in the symmetric group of degree \(n\), \(S_n\), and a character \(\chi\,\) of \(G\), the generalized matrix function of \(A=(A_{ij})\in \mathbb{C}^{n\times n}\) is defined as\N\[\Nd_{\chi}^G(A)=\sum_{\sigma\in G}\chi(g)\prod_{i=1}^n{A_{i\sigma(i)}}.\N\]\NThe permanent and the determinant are two particular cases of this function, respectively, for \(\chi=1,\) the principal character, and \(\chi=e\), the alternating character.\N\NIn the sixties of the last century, \textit{E. H. Lieb} [J. Math. Mech. 16, 127--134 (1966; Zbl 0144.26802)] proposed the Permanent Dominant Conjecture for the normalized generalized matrix function\N\(\bar{d}_{\chi}^G (A):=(1/\chi(e))d_{\chi}^G (A)\) as\N\[\N\bar{d}_{\chi}^G (A)\leq \mathrm{per}(A)\N\]\Nfor any positive semi-definite matrix \(A\in \mathbb{C}^{n\times n}\) and \(\chi\) an irreducible character of \(G\leq S_n.\)\NThis longstanding conjecture remains still open and is revisited in this paper.\N\NAfter stating an identity between the determinant and generalized matrix functions, the author obtains a criterion on positive\Nsemi-definite matrices to affirm the conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture can be generated from any positive semi-definite matrix having no zero in the first column.