带符号初等展开的舒尔正性：团蜘蛛与蜘蛛$S(a,b,2)$

中文摘要 AI 辅助

我们证明了色对称函数的舒尔α-ω引理，它用高阶独立数、高阶团数和色数来界定非零舒尔系数所对应的分拆。随后，我们建立了三个等价的优势匹配准则：矩阵准则、霍尔准则和序理想准则，用于从给定的带符号$e_I$-展开中证明舒尔正性。作为应用，我们对四类基本的3-团蜘蛛的$e$-正性和舒尔正性进行了完整分类。这里$S^{ghk}_{rst}$是通过将一个公共中心分别与$K_r$、$K_s$、$K_t$的一个顶点用内部不交的长度为$g$、$h$、$k$的路径连接而成的图。由此，$S^{000}_{rst}$为舒尔正的充要条件是$r\geq st-1$；每个$S^{100}_{rst}$和$S^{010}_{rst}$均为舒尔正的。当$s=t$时，图$S^{001}_{rst}$是舒尔正的；当$s>t$时，其舒尔正成员分为四个明确的参数区间。我们还引入了路径-团自举法，并用它证明每个蜘蛛$S(a,b,2)$都是舒尔正的。最后，我们证明对于满足$a\geq b\geq2$且$3\nmid b$的蜘蛛$S(a,b,2)$，其为$e$-正的充要条件是$(a,b)\in\{(6,4),(12,4),(9,7)\}$，这推进了关于蜘蛛$S(a,b,2)$的$e$-正性的汤姆猜想的研究。

英文摘要

We prove a Schur alpha-omega lemma for chromatic symmetric functions. It bounds the partitions indexing nonzero Schur coefficients in terms of higher independence numbers, higher clique numbers, and the chromatic number. We then establish three equivalent dominance-matching criteria: matrix, Hall, and order-ideal, that certify Schur positivity from a fixed signed $e_I$-expansion. As applications, we obtain complete classifications of $e$-positivity and Schur positivity for four basic families of $3$-clique-spiders. Here $S^{ghk}_{rst}$ is formed by joining a common center to one vertex of each of $K_r$, $K_s$, and $K_t$ by internally disjoint paths of lengths $g$, $h$, and $k$, respectively. As a result, $S^{000}_{rst}$ is Schur positive exactly when $r\ge st-1$, and every graph $S^{100}_{rst}$ and $S^{010}_{rst}$ is Schur positive. When $s=t$, the graph $S^{001}_{rst}$ is Schur positive; when $s>t$, its Schur-positive members fall into four explicit parameter regimes. We also introduce a path-clique bootstrap and use it to prove that every spider $S(a,b,2)$ is Schur positive. Finally, we prove that the spider $S(a,b,2)$ for $a\ge b\ge2$ with $3\nmid b$ is $e$-positive if and only if $(a,b)\in\{(6,4),(12,4),(9,7)\}$, which advances the study of Tom's conjecture concerning the $e$-positivity of spiders $S(a,b,2)$.

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36 pages, 2 figures, 1 table