Signed permutations and degree-one dot action representations for types B and C

A spline is an assignment of polynomials to the vertices of a graph, where the difference of two polynomials along an edge must belong to the ideal labeling that edge. We consider a ring of splines $\mathcal{M}_{H}$ constructed on a graph whose vertices are the Weyl group $\mathfrak{W}_n$ of signed permutations, and whose edges and edge-ideals are defined using an order ideal $H$ of positive roots. These splines are a module over the polynomial ring in two ways, and a $\mathfrak{W}_n$-module by the dot action. These structures on $\mathcal{M}_{H}$ give rise to the graded left and right dot action representations of $\mathfrak{W}_n$. The left representation is the type B/C generalization of the type A dot action for regular semisimple Hessenberg varieties (and thus, chromatic quasisymmetric functions), and the right representation is the same for corresponding manifolds of isospectral matrices (and thus, unicellular LLT polynomials). This paper gives explicit module generators for the degree-one graded piece of $\mathcal{M}_{H}$ and computes the degree-one piece of the both dot action representations for all $H$ using the combinatorial data of $H$.