循环归纳中的除数格与舒尔正性
The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure
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中文总结 AI 辅助
本文对Sundaram引入的两类对称函数$f_n^T$的舒尔正性进行分类，解决其相关猜想，推导Foulkes坐标准则、主指标剩余不等式等结果，为对称函数研究提供关键结论。
中文摘要 AI 辅助
我们对Sundaram引入的两类对称函数$f_n^T$的舒尔正性进行分类。当$k\ge2$时，$f_n^{\{1,k\}}$恰在$n=k$且$k$为大于2的偶数时不具有舒尔正性；对于$T_k=\{k^i:i\ge0\}$，$f_n^{T_k}$恰在$k$为大于2的偶数且$n=k^a$（其中$a\ge1$）时不具有舒尔正性。在所有例外次数中，$s_{(1^n)}$的系数为$-1$，其余所有舒尔系数均非负。结合Sundaram的乘积蕴含关系，这些分类结果解决了她的猜想1至猜想3。我们还将Sundaram的 plethystic恒等式重写为以除数为索引的坐标形式，并对每个固定次数及全局的非负Foulkes坐标进行分类。若合数$k$整除$n$，即使$f_n^{\{1,k\}}$和$f_n^{T_k}$具有舒尔正性，它们也会有一个负的Foulkes坐标。推广Hou的有界区间论证，我们证明了$n\ge18$时实除数权重的统一准则：若$1\in T$且默比乌斯权重$\boldsymbol{\psi}^T$满足所需界，则对每个奇数$n\ge18$，$f_n^T$均具有舒尔正性；对于偶数$n\ge18$，仅当$n\in T$且$n/2\notin T$时不具有舒尔正性，此时$s_{(1^n)}$是唯一的负舒尔系数，其值为$-1$。最后，对于$k=q^s$（其中$q$为素数，$s\ge2$）、$k\mid n$且$n>k$的情况，我们推导了主指标剩余不等式并对所有等号情形进行分类。对于$n\ge18$，我们还获得了对第一行和第一列长度均不超过$n/2$的分拆的统一定量估计。
英文摘要
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.
