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Dimension formula for the space of relative symmetric polynomials of $$D_n$$ with respect to any irreducible representation. (English) Zbl 1437.05233
Summary: For positive integers $$d$$ and $$n$$, the vector space $$H_d(x_1,x_2,\dots ,x_n)$$ 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 E. Babaei and Y. Zamani [Bull. Iran. Math. Soc. 40, No. 4, 863–874 (2014; Zbl 1338.05271)]. 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 C. F. Fowler et al. [Ramanujan J. 35, No. 2, 205–241 (2014; Zbl 1368.11090)], with the unique supercharacter theory of a group of order 2.
##### MSC:
 05E05 Symmetric functions and generalizations 15A69 Multilinear algebra, tensor calculus 11L03 Trigonometric and exponential sums, general
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##### References:
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