Hook immanantal inequalities for totally nonnegative matrices

Given a weakly decreasing positive integer sequence λ= (λ₁,,λ₎ summing to n, let χ^λ denote the irreducible character of the symmetric group Sₙ indexed by $λ$. This representation has dimension χ^λ(e), where e is the identity element of Sₙ. Let Immχ^λ denote the corresponding irreducible character immanant, the function on n × n matrices A = (ai,j) defined by Immχ^λ(A) := ∑w ∈ Sₙ χ^λ(w) a1,w₁ ⋯ an,wₙ. 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, , 1) satisfy the inequalities $per(A)={Immχ^n(A)}{χⁿ(e)}≥ {Imm_{χn-1,1}(A)}{χn-1,1(e)}≥ {Imm_{χn-2,1,1}(A)}{χn-2,1,1(e)}≥ ⋯ ≥ {Imm_{χ1,,1}(A)}{χ1,,1(e)}=(A)$ whenever $A$ is an $n × n$ Hermitian positive semidefinite matrix. We prove that the same inequalities hold whenever $A$ is an $n × n$ totally nonnegative matrix.