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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:
  Babaei, E.; Zamani, Y.; Shahryari, M., Symmetry classes of polynomials, Commun. Algebra, 44, 1514-1530 (2016) · Zbl 1338.05272  Diaconis, P.; Martin Isaacs, I., Supercharacters and superclasses for algebra groups, Trans. Am. Math. Soc., 360(5), 2359-2363 (2008) · Zbl 1137.20008  Fowler, Christopher F.; Garcia, Stephan Ramon; Karaali, Gizem, Ramanujan sums as supercharacters, The Ramanujan Journal, 35, 2, 205-241 (2013) · Zbl 1368.11090  Hendrickson A, Supercharacter Theories of Finite Cyclic Groups, Ph.D. thesis (2008) (Madison: University of Wisconsin)  Lamar Jonathan P, Lattices of Supercharacter Theories, Mathematics Graduate Theses & Dissertations, 60 (2018) (University of Colorado)  Ramanujan S. On certain trigonometrical sums and their applications in the theory of numbers, in: Collected papers of Srinivasa Ramanujan, pp. 179-199 (2000) (Providence: AMS Chelsea Publ.) (2000), Trans. Cambridge Philos. Soc.22(13) (1918) 259-276  Serre J-P, Linear Representations of Finite Groups (1977) (Springer-Verlag)  Shahryari, M., Relative symmetric polynomials, Linear Algebra Appl., 433, 1410-1421 (2010) · Zbl 1194.05162  Shahryari, M.; Zamani, Y., Symmetry classes of tensors associated with Young subgroups, Asian-Eur. J. Math., 4(1), 179-185 (2011) · Zbl 1211.15034  von Sterneck R D, Sitz. ber. Akad. Wiss. Wien Math. Nat. wiss. Kl. 111(Abt. IIa) (1902) 1567-1601  Zamani, Yousef; Babaei, Esmaeil, THE DIMENSIONS OF CYCLIC SYMMETRY CLASSES OF POLYNOMIALS, Journal of Algebra and Its Applications, 13, 2, 1350085 (2013) · Zbl 1290.05156  Zamani Y and Babaei E, Symmetry classes of polynomials associated with the dicyclic group, Asian-Eur. J. Math.6(3) (2013) Article ID 1350033, 10 pages · Zbl 1277.05168  Zamani, Y.; Babaei, E., Symmetry classes of polynomials associated with the dihedral group, Bull. Iranian Math. Soc., 40(4), 863-874 (2014) · Zbl 1338.05271
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