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arXiv · 2409.00943

Schur-positivity for generalized nets

Ethan Shelburne·Stephanie van Willigenburg
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Abstract

A graph is Schur-positive if its chromatic symmetric function expands nonnegatively in the Schur basis. All claw-free graphs are conjectured to be Schur-positive. We introduce a combinatorial object corresponding to a graph G, called a special rim hook G-tabloid, which is a variation on the special rim hook tabloid. These objects can be employed to compute any Schur coefficient of the chromatic symmetric function of a graph. We construct sign-reversing maps on these special rim hook G-tabloids to obtain a recurrence relation for the Schur coefficients of a family of claw-free graphs called generalized nets, then we prove the entire family is Schur-positive. We subsequently determine an analogous recurrence relation for another, similar family of claw-free graphs. Thus, we demonstrate a new method for proving Schur-positivity of chromatic symmetric functions, which has the potential to be applied to make further progress toward the aforementioned conjecture.

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Ethan Shelburne, Stephanie van Willigenburg. 2024-09-02. Schur-positivity for generalized nets. https://arxiv.org/abs/2409.00943

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