Dimension formula for the space of relative symmetric polynomials of Dn with respect to any irreducible representation

For positive integers d and n, the vector space H-d (x(1), x(2), ..., x(n)) of homogeneous polynomials of degree d is a representation of the symmetric group S-n acting by permutation of variables. Regarding this as a representation for the dihedral subgroup D-n, we prove a formula for the dimension of all the isotypical subrepresentations. Our formula is simpler than the existing one found by Zamani and Babaei (Bull. Iranian Math. Soc. 40(4) (2014) 863-874). By varying the degrees d we compute the generating functions for these dimensions. Further, our formula leads us naturally to a specific supercharacter theory of D-n. It turns out to be a *-product of a specific supercharacter theory studied in depth by Fowler et al. (The Ramanujan Journal (2014)), with the unique supercharacter theory of a group of order 2.