Hyperoctahedral group characters and a type-BC analog of graph coloring

We state combinatorial formulas for hyperoctahedral group (${\mathfrak{B}}_n$) character evaluations of the form $\chi ({{\tilde{C}}_w}^{\text{\hspace{0.25em}⁣}\text{\hspace{0.25em}⁣}BC}\text{\hspace{0.25em}⁣}(1))$, where ${{\tilde{C}}_w}^{\text{\hspace{0.25em}⁣}\text{\hspace{0.25em}⁣}BC}\text{\hspace{0.25em}⁣}(1)\in \mathbb{Z}[{\mathfrak{B}}_n]$ is a type-BC Kazhdan-Lusztig basis element, with $w\in {\mathfrak{B}}_n$ corresponding to simultaneously smooth type-B and C Schubert varieties. We also extend the definition of symmetric group codominance to elements of ${\mathfrak{B}}_n$ and show that for each element $w\in {\mathfrak{B}}_n$ above, there exists a BC-codominant element $v\in {\mathfrak{B}}_n$ satisfying $\chi ({{\tilde{C}}_w}^{\text{\hspace{0.25em}⁣}\text{\hspace{0.25em}⁣}BC}\text{\hspace{0.25em}⁣}(1))=\chi ({{\tilde{C}}_v}^{\text{\hspace{0.25em}⁣}\text{\hspace{0.25em}⁣}BC}\text{\hspace{0.25em}⁣}(1))$ for all ${\mathfrak{B}}_n$-characters $\chi$. Combinatorial structures and maps appearing in these formulas are type-BC extensions of planar networks, unit interval orders, indifference graphs, poset tableaux, and colorings. Using the ring of type-BC symmetric functions, we introduce natural generating functions $Y({{\tilde{C}}_w}^{\text{\hspace{0.25em}⁣}\text{\hspace{0.25em}⁣}BC}\text{\hspace{0.25em}⁣}(1))$ for the above evaluations. These provide a new type-BC analog of Stanley’s chromatic symmetric functions [Adv. Math. 111 (1995) pp. 166-194].