Hook Immanantal Inequalities for Totally Nonnegative Matrices
Abstract
Given a weakly decreasing positive integer sequence λ = (λ1,dotsc,λℓ) summing to n, let χλ denote the irreducible character of the symmetric group Sn indexed by λ. This representation has dimension χλ(e), where e is the identity element of Sn. Let Immχλ denote the corresponding irreducible character immanant, the function on n × n matrices A = (ai,j) defined byImmχλ(A) := negthickspace ∑w ∈ Sn negthickspace χλ(w) hspace.5mm 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, dotsc, 1) satisfy the inequalitiesImmχn(A)/χn(e)≥ (Immχn−1,1(A))/(χn−1,1(e))≥ (Immχ n−2,1,1(A))/(χn−2,1,1(e))≥ ⋯ ≥ Immχ1,dotsc,1(A)/χ1,dotsc,1(e),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.
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BibTeX
@article{skandera2026hook,
title = {{Hook Immanantal Inequalities for Totally Nonnegative Matrices}},
author = {Mark Skandera},
journal = {The Electronic Journal of Combinatorics},
year = {2026},
month = oct,
doi = {10.37236/14684},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/p33i4p23},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Hook Immanantal Inequalities for Totally Nonnegative Matrices
AU - Mark Skandera
PY - 2026
DA - 2026/10/23
JO - The Electronic Journal of Combinatorics
DO - 10.37236/14684
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/p33i4p23
AB - Given a weakly decreasing positive integer sequence λ = (λ_1,dotsc,λ_ℓ) 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 Imm_χ^λ denote the corresponding irreducible character immanant, the function on n × n matrices A = (a_i,j) defined byImm_χ^λ(A) := negthickspace ∑_w ∈ S_n negthickspace χ^λ(w) hspace.5mm a_1,w_1 ⋯ 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 inequalitiesImm_χ^n(A)/χ^n(e)≥ (Imm_χ^n−1,1(A))/(χ^n−1,1(e))≥ (Imm_χ^ n−2,1,1(A))/(χ^n−2,1,1(e))≥ ⋯ ≥ Imm_χ^1,dotsc,1(A)/χ^1,dotsc,1(e),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.
SN - 1077-8926
ER -