Relative symmetric polynomials

Abstract

This article is devoted to the number of non-negative solutions of the linear Diophantine equationwhere a1, . . . , an, and d are positive integers. We obtain a relation between the number of solutions of this equation and characters of the symmetric group, using relative symmetric polynomials. As an application, we give a necessary and sufficient condition for the space of the relative symmetric polynomials to be non-zero.

Relative symmetric polynomials

“…In fact, we will show that Q d is a permutation character of S m , and then we will find its irreducible constituents. Our main tool, in the investigation of Q d , is the notion of relative symmetric polynomials, which is introduced by the author in [3]. Once, we find the irreducible constituents of Q d , we can also obtain a necessary and sufficient condition for vanishing of the space Date: November 9, 2018.…”

“…We need a survey of results about relative symmetric polynomials in this article. For a detailed exposition, one can see [3].…”

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

“…The image of the Reynold's operator will be the space of all symmetric polynomials of degree d. M. Shahryari [3] has introduced the notion of relative symmetric polynomials for any subgroup G ⊂ S n with respect to any irreducible character χ of G.…”

Symmetry Classes of Polynomials

“…For a review of the theory of symmetric polynomials, one can see the book of Macdonald [6]. The relative symmetric polynomials as a generalization of symmetric polynomials are introduced in [10] by Shahryari. In [1,2,12,13], the authors studied the space of relative symmetric polynomials (symmetry class of polynomials) with respect to the irreducible characters of certain groups. In this article, we investigate an embedding of the symmetry classes of polynomials into the ordinary symmetry classes of tensors, and using it we obtain some results concerning nonvanishing of the symmetry classes of polynomials.…”

“…Also we study the symmetry classes of polynomials with respect to the irreducible characters of the symmetric group S m and the alternating group A m . We first give a review of this notion (for more details, see [10]). …”