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

We state combinatorial formulas for hyperoctahedral group ( 𝔅 n ) character evaluations of the form χ ( C ˜ w BC ( 1 ) ) where C ˜ w BC ( 1 ) ∈ ℤ [ 𝔅 n ] is a type- BC Kazhdan–Lusztig basis element, with w ∈ 𝔅 n corresponding to simultaneously smooth type- B and C Schubert varieties. We also extend the definition of symmetric group codominance to elements of 𝔅 n and show that for each element w ∈ 𝔅 n as above, there exists a BC -codominant element v ∈ 𝔅 n satisfying χ ( C ˜ w BC ( 1 ) ) = χ ( C ˜ v BC ( 1 ) ) for all 𝔅 n -characters χ . 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 ( C ˜ w BC ( 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].