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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.
05E05 Symmetric functions and generalizations
15A69 Multilinear algebra, tensor calculus
11L03 Trigonometric and exponential sums, general
Full Text: DOI
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