Dimension formula for the space of relative symmetric polynomials of $$\varvec{D_n}$$ with respect to any irreducible representation

Abstract

For positive integers d and n, the vector space $$H_d(x_1,x_2,\ldots ,x_n)$$ 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.