Hook immanantal inequalities for totally nonnegative matrices
Mark Skandera
TL;DR
This work extends Merris–Heyfron’s hook-immanant inequalities from Hermitian positive semidefinite matrices to totally nonnegative matrices, showing that the chain $\mathrm{per}(A) \geq \mathrm{Imm}_{\chi^{n-1,1}}(A)/\chi^{n-1,1}(e) \geq \cdots \geq \mathrm{Imm}_{\chi^{1,\dots,1}}(A)/\chi^{1,\dots,1}(e)$ holds for all $n\times n$ totally nonnegative matrices. It develops a dual approach: a constructive planar-network argument giving a direct hook-immanant inequality, and a second proof via a nonnegative decomposition into $\theta^{\ell}$-immanants, alongside a rich symmetric-function and poset framework that links immanants to chromatic symmetric functions and $P$-tableaux. The paper leverages the Frobenius correspondence between $\mathcal{T}_n$ and $\Lambda_n$, the SkanCCS methodology, and planar-network theory to interpret and prove positivity results for induced sign/trivial and hook immanants, while outlining open problems for general $\lambda,\mu$ and $q$-analogs. Overall, the results advance the understanding of positivity and dominance among immanants in the total nonnegativity setting and connect representation-theoretic, combinatorial, and network-theoretic methods with implications for further research.
Abstract
Given a weakly decreasing positive integer sequence $λ= (λ_1,\dotsc,λ_\ell)$ summing to $n$, let $χ^λ$ denote the irreducible character of the symmetric group $S_n$ indexed by $λ$. This representation has dimension $χ^λ(e)$, where $e$ is the identity element of $S_n$. Let $\mathrm{Imm}_{χ^λ}$ denote the corresponding irreducible character immanant, the function on $n \times n$ matrices $A = (a_{i,j})$ defined by $\mathrm{Imm}_{χ^λ}(A) := \sum_{w \in S_n} χ^λ(w) a_{1,w_1} \cdots a_{n,w_n}$. 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 $\mathrm{per}(A)=\frac{\mathrm{Imm}_{χ^n}(A)}{χ^{n}(e)}\geq \frac{\mathrm{Imm}_{χ^{n-1,1}}(A)}{χ^{n-1,1}(e)}\geq \frac{\mathrm{Imm}_{χ^{ n-2,1,1}}(A)}{χ^{n-2,1,1}(e)}\geq \cdots \geq \frac{\mathrm{Imm}_{χ^{1,\dotsc,1}}(A)}{χ^{1,\dotsc,1}(e)}=\det(A)$ 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.