Hook immanantal inequalities for totally nonnegative matrices

Abstract

Given a weakly decreasing positive integer sequence $\lambda = (\lambda_1,\dotsc,\lambda_\ell)$ 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 $\mathrm{Imm}_{\chi^\lambda}$ denote the corresponding irreducible character immanant, the function on $n \times n$ matrices $A = (a_{i,j})$ defined by $\mathrm{Imm}_{\chi^\lambda}(A) := \sum_{w \in S_n} \chi^\lambda(w) a_{1,w_1} \cdots a_{n,w_n}$. 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 $(k, 1, \dotsc, 1)$ satisfy the inequalities $\mathrm{per}(A)=\frac{\mathrm{Imm}_{\chi^n}(A)}{\chi^{n}(e)}\geq \frac{\mathrm{Imm}_{\chi^{n-1,1}}(A)}{\chi^{n-1,1}(e)}\geq \frac{\mathrm{Imm}_{\chi^{ n-2,1,1}}(A)}{\chi^{n-2,1,1}(e)}\geq \cdots \geq \frac{\mathrm{Imm}_{\chi^{1,\dotsc,1}}(A)}{\chi^{1,\dotsc,1}(e)}=\det(A)$ 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.