The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure

Abstract

We study the Schur-positive cone in $\Rspace_{n,\mathbb R}\coloneqq \operatorname{span}_{\mathbb R}\{p_d^{n/d}:d\mid n\}$ through its basis $Q_{n,d}\coloneqq\ell_{n/d}^{(1)}[p_d]$, where $\ell_m^{(1)}$ 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 $\sum_{d\in J}Q_{n,d}$ is Schur-positive exactly when $1\in J$ and, for even $n$, $n\in J$ implies $n/2\in J$. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function $\sum_{d\mid n}c_d(n/d)^2p_d^{n/d}$, where $c_d(r)$ is the Ramanujan sum, has a positive coefficient of $s_λ$ for every $n\ge1$ and $λ\vdash n$, except when $n\equiv2\pmod4$ 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 $n\ge18$, 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.