The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure
Abstract
We study the Schur-positive cone in Rspacen,ℝcoloneqq spanℝ{pdn/d:d| n} through its basis Qn,dcoloneqqℓn/d(1)[pd], where ℓm(1) is the Frobenius characteristic of the representation induced to Sm from a faithful linear character of the subgroup generated by an m-cycle; brackets denote plethysm. A Boolean Q-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean Q-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set J of divisors of n, the sum ∑d∈ JQn,d is Schur-positive exactly when 1∈ J and, for even n, n∈ J implies n/2∈ J. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function ∑d| ncd(n/d)2pdn/d, where cd(r) is the Ramanujan sum, has a positive coefficient of sλ for every n≥1 and λ⊢ n, except when n≡2pmod4 and λ=(1n), in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its Q-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with Qn,1-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For n≥18, we prove that setting the coefficient of s(n) or s(1n) equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with Qn,1-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.
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BibTeX
@misc{oh2026cyclicinduction,
title = {{The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure}},
author = {Young-Tak Oh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29757},
eprint = {2608.29757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29757},
}
RIS
TY - UNPB
TI - The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure
AU - Young-Tak Oh
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.29757
UR - https://arxiv.org/abs/2608.29757
AB - We study the Schur-positive cone in Rspace_n,ℝcoloneqq span_ℝ{p_d^n/d:d| n} through its basis Q_n,dcoloneqqℓ_n/d^(1)[p_d], where ℓ_m^(1) is the Frobenius characteristic of the representation induced to S_m from a faithful linear character of the subgroup generated by an m-cycle; brackets denote plethysm. A Boolean Q-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean Q-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set J of divisors of n, the sum ∑_d∈ JQ_n,d is Schur-positive exactly when 1∈ J and, for even n, n∈ J implies n/2∈ J. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function ∑_d| nc_d(n/d)^2p_d^n/d, where c_d(r) is the Ramanujan sum, has a positive coefficient of s_λ for every n≥1 and λ⊢ n, except when n≡2pmod4 and λ=(1^n), in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its Q-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with Q_n,1-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For n≥18, we prove that setting the coefficient of s_(n) or s_(1^n) equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with Q_n,1-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.
ER -