Schur positivity from signed elementary expansions: clique-spiders and spiders $S(a,b,2)$

Abstract

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 $S(a,b,2)$.