The Dimensions of Cyclic Symmetry Classes of Polynomials

Abstract

The dimensions of the symmetry classes of polynomials with respect to a certain cyclic subgroup of Sm generated by an m-cycle are explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed in terms of the Euler ϕ-function and the Möbius function for some special cases.

Dimension formula for the space of relative symmetric polynomials of $$\varvec{D_n}$$ with respect to any irreducible representation

“…In later papers Babaei, Zamani and Shahryari found the dimension of the space of relative invariants for S n and its subgroup A n [3] and for Young subgroup [4] In a series of papers Babaei and Zamani have given corresponding formula for the cyclic group in [7], for the dicyclic group in [8] and for the dihedral group D n [9].…”

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.…”