Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants
Abstract
Haiman conjectured that for every partition λ⊢ n and every w∈ Sn the polynomial cλ,w(q), a monomial character of the Hecke algebra evaluated at the Kazhdan-Lusztig basis, has non-negative integer coefficients and is symmetric and unimodal, and he verified this by computer for n≤7. We verify it for every n≤10. A stability lemma reduces the statement to n=10, which we settle by an exact computation that stores only the 1,252,173,030 pairs (x,w) in S10 with x maximal in its double coset for the descent sets of w, and takes the traces of the standard basis from the class polynomials of Geck and Pfeiffer. Every stored Kazhdan-Lusztig polynomial agrees with du Cloux's Coxeter3, and the coefficients pass several independent checks, including a comparison at all 16,796 codominant w with chromatic quasisymmetric functions computed from colourings. Consequently, for n≤10 every monomial immanant is a non-negative integer combination of Kazhdan-Lusztig immanants, Stembridge's monomial immanant conjecture holds, and a class-function immanant is non-negative on totally non-negative matrices if and only if it is non-negative at the block matrices Jμ. We use this to describe the order of normalised irreducible-character immanants on totally non-negative matrices for n≤10, and to prove Pate's inequalities there. On the full span of permutation monomials we give a Newton-polytope criterion for the coefficient of Immw to be non-negative throughout the totally non-negative cone, show that for n=4,5 this happens exactly when Xw is smooth, and check by exact computation for n≤10 the proof of this for all n given in a companion paper.
Related
- Hook monomial immanants are Kazhdan-Lusztig positive
- Relative Rouquier linearity and inverse hybrid Kazhdan-Lusztig positivity
- Hybrid Kazhdan-Lusztig basis and parabolic induction
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
- On the nonnegativity of monomial immanants for hook partitions
BibTeX
@misc{mcnulty2026haimans,
title = {{Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants}},
author = {Keith McNulty},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41555},
eprint = {2610.41555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41555},
}
RIS
TY - UNPB
TI - Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants
AU - Keith McNulty
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41555
UR - https://arxiv.org/abs/2610.41555
AB - Haiman conjectured that for every partition λ⊢ n and every w∈ S_n the polynomial c_λ,w(q), a monomial character of the Hecke algebra evaluated at the Kazhdan-Lusztig basis, has non-negative integer coefficients and is symmetric and unimodal, and he verified this by computer for n≤7. We verify it for every n≤10. A stability lemma reduces the statement to n=10, which we settle by an exact computation that stores only the 1,252,173,030 pairs (x,w) in S_10 with x maximal in its double coset for the descent sets of w, and takes the traces of the standard basis from the class polynomials of Geck and Pfeiffer. Every stored Kazhdan-Lusztig polynomial agrees with du Cloux's Coxeter3, and the coefficients pass several independent checks, including a comparison at all 16,796 codominant w with chromatic quasisymmetric functions computed from colourings. Consequently, for n≤10 every monomial immanant is a non-negative integer combination of Kazhdan-Lusztig immanants, Stembridge's monomial immanant conjecture holds, and a class-function immanant is non-negative on totally non-negative matrices if and only if it is non-negative at the block matrices J_μ. We use this to describe the order of normalised irreducible-character immanants on totally non-negative matrices for n≤10, and to prove Pate's inequalities there. On the full span of permutation monomials we give a Newton-polytope criterion for the coefficient of Imm_w to be non-negative throughout the totally non-negative cone, show that for n=4,5 this happens exactly when X_w is smooth, and check by exact computation for n≤10 the proof of this for all n given in a companion paper.
ER -