Symmetry Classes of Polynomials

Abstract

Let G be a subgroup of S m and suppose is an irreducible complex character of G. Let H d G be the symmetry class of polynomials of degree d with respect to G and . Let V be an d + 1 -dimensional inner product space over and V G be the symmetry class of tensors associated with G and . A monomorphism H d G → V G is given and it is used to obtain necessary and sufficient conditions for nonvanishing H d G . The nonexistence of o-basis of H d S m 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.

Generalized symmetry classes of tensors

“…, the symmetry class of tensors associated with G and the irreducible character λ of G corresponding to the representation Λ (see [4], [5], [9], [10], [12], [13], [14]). Recently, the other types of symmetry classes have been studied by several authors (see [1], [2], [3], [7], [11], [15], [16]).…”

Representations of the general linear group over symmetry classes of polynomials

“…In [2], Zamani and Babaei studied symmetry classes of polynomials with respect to irreducible characters of the direct product of permutation groups. In [1], [3], [9], [10], they computed the dimensions of symmetry classes of polynomials with respect to irreducible characters of dihedral, symmetric, dicyclic and cyclic groups, respectively. Also, they discussed the existence of an o-basis for these classes.…”