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

Abstract

Determining when the chromatic symmetric function of a graph is Schur positive is a central problem in algebraic combinatorics. Elementary expansions can help answer this question, but negative coefficients obstruct a direct positivity argument. We develop methods for proving Schur positivity from signed elementary expansions and apply them to chromatic symmetric functions of clique-spiders and three-legged spiders. We first give the Schur alpha-omega lemma, which bounds Schur support. We then give four equivalent sufficient criteria for Schur positivity from a fixed composition-indexed elementary expansion. The matrix, order-ideal, negative Hall, and positive Hall formulations treat positive coefficients as capacities and negative coefficients as demands, with admissible assignments governed by dominance order. These criteria provide certificates of Schur positivity especially when elementary coefficients are negative. For the four basic families of $3$-clique-spiders with total leg length at most one, we obtain complete classifications of both $e$-positivity and Schur positivity. These classifications show how the position of a single leg edge changes positivity. Using selected clique-spider cases and a path-clique bootstrap, we prove that all spiders of the form $S(a,b,2)$ are Schur positive.

26 pages, 2 figures, 1 table