Hook immanantal inequalities for totally nonnegative matrices

Abstract

Given a weakly decreasing positive integer sequence $\hat{I}»=(\hat{I}{»}_1,\dots ,\hat{I}{»}_{\ell })$ summing to $n$, let $\ddot{I}{‡}^{\hat{I}}»$ denote the irreducible character of the symmetric group $S_n$ indexed by $\hat{I}»$. This representation has dimension $\ddot{I}{‡}^{\hat{I}}»(e)$, where $e$ is the identity element of $S_n$. Let ${\mathrm{Imm}}_{\ddot{I}{‡}^{\hat{I}}»}$ denote the corresponding irreducible character immanant, the function on $n\times n$ matrices $A=(a_{i,j})$ defined by ${\mathrm{Imm}}_{\ddot{I}{‡}^{\hat{I}}»}(A):={\sum }_{w\in S_n}\ddot{I}{‡}^{\hat{I}}»(w)a_{1,w_1}\cdots a_{n,w_n}$. 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 $\mathrm{per}(A)=\frac{{\mathrm{Imm}}_{\ddot{I}{‡}^n}(A)}{\ddot{I}{‡}^n(e)}\ge \frac{{\mathrm{Imm}}_{\ddot{I}{‡}^{n-1,1}}(A)}{\ddot{I}{‡}^{n-1,1}(e)}\ge \frac{{\mathrm{Imm}}_{\ddot{I}{‡}^{n-2,1,1}}(A)}{\ddot{I}{‡}^{n-2,1,1}(e)}\ge \cdots \ge \frac{{\mathrm{Imm}}_{\ddot{I}{‡}^{1,\dots ,1}}(A)}{\ddot{I}{‡}^{1,\dots ,1}(e)}=\det (A)$ 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.

17 pages