## Abstract

Let G be a subgroup of S_{m} and suppose χ is an irreducible complex character of G. Let H_{d}(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 V_{χ}(G) be the symmetry class of tensors associated with G and χ. A monomorphism H_{d}(G, χ) → V_{χ}(G) is given and it is used to obtain necessary and sufficient conditions for nonvanishing H_{d}(G, χ). The nonexistence of o-basis of H_{d}(S_{m}, χ^{π}) for a certain class of irreducible characters of S_{m} is concluded. The dimensions of symmetry classes of polynomials with respect to the irreducible characters of S_{m} and A_{m} are computed.

Original language | English |
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Pages (from-to) | 1514-1530 |

Number of pages | 17 |

Journal | Communications in Algebra |

Volume | 44 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2 2016 |

Externally published | Yes |

## Keywords

- Alternating group
- Irreducible characters
- Orthogonal basis
- Symmetric group
- Symmetry class of polynomials
- Symmetry class of tensors

## ASJC Scopus subject areas

- Algebra and Number Theory