Hook immanantal inequalities for totally nonnegative matrices

Given a weakly decreasing positive integer sequence $\lambda =({\lambda }_1,\dots ,{\lambda }_{\mathrm{\ell }})$ summing to $n$, let ${\chi }^{\lambda }$ denote the irreducible character of the symmetric group $S_n$ indexed by $\lambda$. This representation has dimension ${\chi }^{\lambda }(e)$, where $e$ is the identity element of $S_n$. Let ${\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^{\lambda }}$ denote the corresponding irreducible character immanant, the function on $n\times n$ matrices $A=(a_{i,j})$ defined by ${\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^{\lambda }}(A):={\sum }_{w\in S_n}{\chi }^{\lambda }(w)a_{1,w_1}\cdots a_{n,w_n}$. Merris conjectured [Linear Multilinear Algebra 14 (1983) pp. 21–35] and Heyfron proved [Linear Multilinear Algebra 24 (1988) pp. 65–78] that irreducible character immanants indexed by ``hook’’ sequences $(k,1,\dots ,1)$ satisfy the inequalities $\mathrm{p}\mathrm{e}\mathrm{r}(A)=\frac{{\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^n}(A)}{{\chi }^n(e)}\ge \frac{{\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^{n-1,1}}(A)}{{\chi }^{n-1,1}(e)}\ge \frac{{\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^{n-2,1,1}}(A)}{{\chi }^{n-2,1,1}(e)}\ge \cdots \ge \frac{{\mathrm{I}\mathrm{m}\mathrm{m}}_{{\chi }^{1,\dots ,1}}(A)}{{\chi }^{1,\dots ,1}(e)}=\det (A)$ whenever $A$ is an $n\times n$ Hermitian positive semidefinite matrix. We prove that the same inequalities hold whenever $A$ is an $n\times n$ totally nonnegative matrix.