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

Abstract

We state combinatorial formulas for hyperoctahedral group ($\mathfrak B_n$) character evaluations of the form $\chi( {{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1))$, where ${{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1) \in \Bbb Z[\mathfrak B_n]$ 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 $\chi( {{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1)) = \chi( {{\widetilde C}_v}^{\negthickspace\negthickspace BC}\negthickspace(1))$ 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 $Y( {{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1))$ for the above evaluations. These provide a new type-BC analog of Stanley's chromatic symmetric functions [Adv. Math. 111 (1995) pp. 166-194].