Combinatorial interpretation of dimensions of Hecke algebras H(S_n, D_n, χ)

Develop a combinatorial interpretation for the vector-space dimensions of the Hecke algebras H(S_n, D_n, χ) associated with the symmetric group S_n, its dihedral subgroup D_n, and a character χ of D_n; specify the combinatorial objects or statistics whose enumeration equals dim H(S_n, D_n, χ) for each χ ∈ Irr(D_n).

Background

The authors define Hecke algebras H(S_n, D_n, χ) attached to characters of the dihedral subgroup and derive consequences about their representation counts via positivity of branching coefficients, showing the number of irreducible representations is close to the number of partitions of n.

They then pose a direct question seeking a combinatorial model that explains the dimensions of these Hecke algebras, aiming to relate algebraic structure to explicit combinatorial enumeration.