On the chromatic symmetric homology for star graphs

Abstract: The chromatic symmetric function $X_G$ is a power series that encodes the proper colorings of a graph $G$ by assigning a variable to each color and a monomial to each coloring such that the power of a variable in a monomial is the number of times the corresponding color is used in the corresponding coloring. The chromatic symmetric homology $H_(G)$ is a doubly graded family of $\mathbb{C}[\mathfrak{S}n]$-modules that was defined by Sazdanovi\'c and Yip (2018) as a categorification of $X_G$. Chandler, Sazdanovi\'c, Stella, and Yip (2023) proved that $H(G)$ is a strictly stronger graph invariant than $X_G$, and they also computed or conjectured formulas for it in a number of special cases. We prove and extend some of their conjectured formulas for the case of star graphs, where one central vertex is connected to all other vertices and no other pairs of vertices are connected.