Abstract
Let G be a subgroup of Sm and suppose χ is an irreducible complex character of G. Let Hd(G, χ) be the symmetry class of polynomials of degree d with respect to G and χ. Let V be an (d + 1)-dimensional inner product space over ℂ and Vχ(G) be the symmetry class of tensors associated with G and χ. A monomorphism Hd(G, χ) → Vχ(G) is given and it is used to obtain necessary and sufficient conditions for nonvanishing Hd(G, χ). The nonexistence of o-basis of Hd(Sm, χπ) for a certain class of irreducible characters of Sm is concluded. The dimensions of symmetry classes of polynomials with respect to the irreducible characters of Sm and Am are computed.
2010 Mathematics Subject Classification:
ACKNOWLEDGMENTS
The authors would like to thank Dr. Kazem Ghanbari for the careful reading of manuscript.